1 Introduction↩︎

We list some notation and make some conventions that will be used throughout the paper.

1. For \(\epsilon>0\) and \(\mathbf{x} \in \mathbb{R}^n\), we let \(\mathbb{B}_\epsilon^n(\mathbf{x})\) denote the ball of radius \(\epsilon\) in \(\mathbb{R}^n\) centered at \(\mathbf{x}\). If \(\epsilon=1\), we often abbreviate \(\mathbb{B}_1^n(\mathbf{x})\) to \(\mathbb{B}^n(\mathbf{x})\); if \(\mathbf{x}=0\), then we often abbreviate \(\mathbb{B}_\epsilon^n(\mathbf{x})\) to \(\mathbb{B}_\epsilon^n\).

2. To simplify our discussions, we always assume that phase functions \(\phi(x, t; \xi)\) are analytic and \(x\in \R^2, t\in \R, \xi\in \R^2\).

3. For a given phase function \(\phi(x, t; \xi)\), we always work with \((x, t)\in \B^{(3)}_{\epsilon_{\phi}}\) and \(\xi\in \B^{(2)}_{\epsilon_{\phi}}\), where \(\epsilon_{\phi}>0\) is a sufficiently small constant depending only on \(\phi\).

4. Manifolds that appear in the current paper are always assumed to be analytic and three dimensional, unless otherwise stated. Analyticity is used to get rid of functions whose Taylor coefficients are all zero but the function itself is not identically zero.

5. We use \(f\equiv 0\) to mean that the function \(f\) vanishes everywhere, and \(f\equiv g\) to mean that these two functions are equal everywhere.

6. We largely follow the notation in the paper [2]. Moreover, for readers not used to curvature tensor calculations, it may be helpful to take a look at Subsection 3.1 and Subsection 5.1 in [2] for some preliminary calculations (we will also recall them whenever they are applied).

7. We usually use \(\mathcal{R}\) for the Riemannian curvature tensor, unless otherwise defined.

1.1 Hörmander-type oscillatory integrals↩︎

Let \(\phi(x, t; \xi)\) be a phase function satisfying the standard Hörmander non-degeneracy condition (see for instance (H1) and (H2) in Guth, Hickman and Iliopoulou [5]). Here \(x\in \R^2, t\in \R, \xi\in \R^2\). We are interested in studying Hörmander-type oscillatory integral operators \[T_N^{(\phi)} f(x, t):= \int_{\R^2} e^{iN \phi(x, t; \xi)} a(x, t; \xi) f(\xi) d\xi,\] where \(N\in \R\) is a large real number, and \(a(x, t; \xi)\) is a smooth function supported in a small neighborhood of the origin. More precisely, we will pick \(\epsilon_{\phi}>0\) to be a sufficiently small constant depending on \(\phi\), and assume that \(a(x, t; \xi)\) is supported inside \[\B^{(2)}_{\epsilon_{\phi}}\times \B^{(1)}_{\epsilon_{\phi}}\times \B^{(2)}_{\epsilon_{\phi}}.\] Here we use \(\B^{(n)}_{\epsilon}\) to denote the ball in \(\R^n\) of radius \(\epsilon\) centered at the origin. The estimates we are interested in proving are of the form \[\label{260503e1952} \Norm{ T_N^{(\phi)} f }_{L^p(\R^3)}\lesim_{\phi, a, p, \epsilon} N^{-\frac{3}{p}+\epsilon} \|f\|_{L^p(\R^2)},\tag{1}\] for every \(\epsilon>0\) and every \(N\ge 1\), and for a range of exponents \(p\) that is as large as possible.
Hörmander [6] and Bourgain [7] observed that to study the estimate 1 , one can apply elementary changes of variables, and only need to consider phase functions \(\phi\) of the form \[\phi(x, t; \xi)= \inn{x}{\xi}+ t\inn{A\xi}{\xi}+ O\pnorm{ |t||\xi|^3+ |\bfx|^2 |\xi|^2 }, \;\;\bfx:=(x, t),\] for some non-degenerate \(2\times 2\) matrix \(A\); these are called normal forms at the origin. If the matrix \(A\) is positive definite, we say that \(\phi\) is elliptic.
The geometry of characteristic curves (see the definition below) plays a central role in the study of Hörmander-type operators.

To study finer properties for the oscillatory integral operator \(T_N^{(\phi)}\) and dimensions of \(\phi\)-Kakeya sets, Bourgain [7] introduced the following notion, which was called Bourgain’s condition in Guo, Wang and Zhang [8].

Examples for which Bourgain’s condition holds include the Fourier extension operator, the Bochner-Riesz operator, the Schrödinger propagator, the resolvent of the standard Laplace operator in the Euclidean space, and so on. We refer to [8] for a detailed discussion on the Bourgain’s condition for these operators and more other operators.

In a certain sense, operators satisfying Bourgain’s condition admit certain “best possible" behavior. For instance, it is proven in [8] that operators with Bourgain’s condition satisfy the same polynomial Wolff axioms as the ones for the standard Fourier extension operators. Moreover, Nadjimzadah [9] proved that for any Hörmander-type oscillatory integral operator satisfying Bourgain’s condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture.
After Bourgain’s work [7], Minicozzi and Sogge [10], Sogge [1], Bourgain and Guth [11], Guth, Hickman and Iliopoulou [5] and Hickman and Iliopoulou [12] studied more Hörmander-type operators that have less favorable properties compared with those satisfying Bourgain’s condition; we also refer to the references therein for more results. Recently, [8], [2] and [3], inspired by the previously mentioned works, tried to push this research direction further; in particular, they tried to classify all Hörmander-type operators. While doing so, some interesting applications (say [13]) were found.
To describe these developments, we need to introduce more notation. For a fixed \(x_0\in\R^2\), let \(X(t; x_0; \xi)\) be the unique solution to \[\label{260502e1951} \nabla_{\xi} \phi(X(t; x_0; \xi), t; \xi)= \nabla_{\xi} \phi(x_0, 0; \xi), \;\forall t.\tag{2}\] In other words, the characteristic curve \(\Gamma_{\xi}(x_0)\) is given by the trajectory of \[(X(t; x_0; \xi), t).\] We introduce the notation \[\label{260504e19510} W_{ij}(t; x_0; \xi):= (\partial_{\xi_i}\partial_{\xi_j}\phi)( X(t; x_0; \xi), t; \xi )- (\partial_{\xi_i}\partial_{\xi_j}\phi)( x_0, 0; \xi ).\tag{3}\] Moreover, define \[\mc{W}(t; x_0; \xi):= \det \begin{bmatrix} W_{11}(t; x_0; \xi), & W_{12}(t; x_0; \xi)\\ W_{21}(t; x_0; \xi), & W_{22}(t; x_0; \xi) \end{bmatrix}.\] Consider the \(4\times \infty\) matrix \[\label{260309e1954} \mathfrak{W}(x_0; \xi):= \begin{bmatrix} \frac{\partial}{\partial t} \mc{W}(0; x_0; \xi), & \frac{\partial^2}{\partial t^2} \mc{W}(0; x_0; \xi), & \dots, & \frac{\partial^k}{\partial t^k} \mc{W}(0; x_0; \xi), & \dots\\ \frac{\partial}{\partial t} W_{11}(0; x_0; \xi), & \frac{\partial^2}{\partial t^2} W_{11}(0; x_0; \xi), & \dots, & \frac{\partial^k}{\partial t^k} W_{11}(0; x_0; \xi), & \dots\\ \frac{\partial}{\partial t} W_{12}(0; x_0; \xi), & \frac{\partial^2}{\partial t^2} W_{12}(0; x_0; \xi), & \dots, & \frac{\partial^k}{\partial t^k} W_{12}(0; x_0; \xi), & \dots\\ \frac{\partial}{\partial t} W_{22}(0; x_0; \xi), & \frac{\partial^2}{\partial t^2} W_{22}(0; x_0; \xi), & \dots, & \frac{\partial^k}{\partial t^k} W_{22}(0; x_0; \xi), & \dots \end{bmatrix}\tag{4}\] Let \(\mathfrak{W}^-(x_0; \xi)\) be the \(3\times \infty\) matrix consisting of the last three rows of \(\mathfrak{W}(x_0; \xi)\). Moreover, let \(\mathfrak{W}_{k}(x_0; \xi)\) and \(\mathfrak{W}^-_{k}(x_0; \xi)\) be the first \(k\) columns of \(\mathfrak{W}(x_0; \xi)\) and \(\mathfrak{W}^-(x_0; \xi)\), respectively.

It is elementary to see that the rank of \(\mathfrak{W}\) is at least two at any given point. To see why the first row in 4 plays a special role in Definition 3, let us compare the example of the standard Fourier extension operator, \[\phi_1(x, t; \xi)= \inn{x}{\xi}+ \frac{1}{2} t|\xi|^2,\] with Bourgain’s example in [7] \[\phi_2(x, t; \xi)= \inn{x}{\xi}+ t\xi_1 \xi_2+ \frac{1}{2} t^2 \xi_1^2,\] where \(x=(x_1, x_2), \xi=(\xi_1, \xi_2)\). Elementary calculation shows that the matrix \(\mathfrak{W}\) always has rank two, for both \(\phi_1\) and \(\phi_2\). However, we expect 1 to hold for all \(p\ge 3\) (the Fourier restriction conjecture) for the phase function \(\phi_1\), while Bourgain [7] showed that 1 fails for all \(p<4\) for the phase function \(\phi_2\).

Let us also point out here that Definition 3 already implicitly appeared in [3], and earlier forms of this definition can be found in Bourgain [7] and Sogge [1].

The case \(r=4\) of Theorem 1 was contained in [2]; the cases \(r=2, 3\) were not stated in [2] but essentially the same argument¹ works, and therefore we will not repeat the proof. For the case \(r=2\), [8] already obtained a range better than ?? .

Theorem 1 may not hold anymore if the rank-\(r\) condition is replaced by rank-\((\le r)\) condition, and this was shown already in Minicozzi and Sogge [10] and Sogge, Xi and Xu [14]. Moreover, Remark 5.4 in [3] provided an example showing why the key ingredient in the argument in [2], which is the polynomial Wolff axiom (see [2]), fails to work under the rank-\((\le r)\) condition.
Theorem 1 concerns Hörmander-type oscillatory integrals. If we let \(E_{\phi}\subset \R^3\) be a \(\phi\)-Kakeya set, then one can essentially follow the argument in Hickman, Rogers and Zhang [15], as was done in [16], and obtain under the same assumption as in Theorem 1 that \[\label{260518e19516} \dim(E_{\phi})\ge \frac{4k-1}{2k-1}.\tag{5}\] Note that when \(k=4\) (which forces \(r=2, 3, 4\)), this is equal to \(2+ 1/7\); when \(k=3\) (which forces \(r=2, 3\)), this is equal to \(2+1/5\); when \(k=2\) (which forces \(r=2\)), this is equal to \(2+1/3\).

1.2 Carleson-Sjölin operators↩︎

We recall the setup of Carleson-Sjölin operators on manifolds, in Subsection 1.3 in [2]. We work with a smooth metric \(\{g_{ij}(\bfx)\}_{1\le i, j\le 3}\) defined on a small neighborhood of the origin \(0\in \R^3\). The manifold is denoted by \(\mc{M}\). Let \(\varepsilon_{\mc{M}}>0\) be a sufficiently small constant depending on \(\mc{M}\). Take \(\varepsilon\le \varepsilon_{\mc{M}}/10\) and let \(\gamma: (-2\varepsilon, 2\varepsilon)\to \mc{M}\) be a geodesic with unit speed. Without loss of generality, we make a normalization so that \[\label{260502e1955} \gamma(0)=(0, 0, 0), \;\;\gamma(s)=(0, 0, s), \forall s.\tag{6}\] Consider the (reduced) Carleson-Sjölin operator, given by \[\label{260518e19518} R_{\varepsilon, N} f(\bfx):= \int_{ |y|\le \varepsilon/10 } e^{ i N \dist(\bfx, (y, \varepsilon)) } a(\bfx) f(y) d y,\tag{7}\] where \(\bfx=(x_1, x_2, x_3)\), \(y=(y_1, y_2)\), and \(a\) is a smooth bump function supported on \(|\bfx|\le \varepsilon/10\). It is elementary (see Subsection 5.2 in [2]) that \[\label{260603e19521zzz} \dist_{\epsilon}(\bfx, y):= \dist(\bfx, (y, \epsilon))\tag{8}\] satisfies Hörmander’s non-degeneracy condition. Take \[\label{260601e19520} \phi(\bfx; y):= \dist_{\epsilon}(\bfx, y).\tag{9}\] We say that \(E\subset \mathcal{M}\) is a Kakeya set on the manifold \(\mathcal{M}\) if it is a \(\phi\)-Kakeya set.
It is proven in [2] that characteristic curves for the phase function are all given by geodesics. More precisely, let \(\gamma\) be a geodesic passing through \((y, \epsilon)\). Then \(\nabla_y \phi(\bfx; y)\) stays constant when \(\bfx\) moves along \(\gamma\). This property explains the geometric meaning of Kakeya sets on manifolds.

1.3 Various curvatures↩︎

Chaotic curvatures of order \(\le 1\) along geodesics were introduced by Sogge [1] when he was studying Kakeya problems on Riemannian manifolds. In Sogge’s paper [1], chaotic curvature of order \(\le 1\) was simply called chaotic curvature. Moreover, he used the name “variably curved manifold" if a manifold \(\mc{M}\) satisfies the chaotic curvature condition of order \(\le 1\) (along every geodesic). Definition 4 generalizes Sogge’s chaotic curvature from order \(\le 1\) to order \(\le k\) for every \(k\).
Let us also remark here that in [1], Sogge studied Kakeya problems not only on manifolds satisfying the chaotic curvature condition, but also on manifolds (of dimension three) that have constant sectional curvature. Later, Xi [17] extended Sogge’s result for three dimensional manifolds of constant sectional curvature to every higher dimension.

Manifolds of constant sectional curvature admit certain “best" possible behavior in the study of Kakeya problems. For instance, it was proven in [8] and [2] that the distance function \(\dist_{\epsilon}(\bfx, y)\) (given in 8 ) satisfies Bourgain’s condition for every \(\epsilon, \bfx\) and \(y\) if and only if the manifold has constant sectional curvature (see also Corollary 1 below).

On the other hand, manifolds satisfying the chaotic curvature condition have “least possible" symmetries², and this explains the name”chaotic" by [1]. At the end of [1], Sogge proposed to study Kakeya problems on general (locally) symmetric spaces, which, in terms of symmetries, lie between manifolds satisfying the chaotic curvature condition and manifolds of constant sectional curvature. We will see that this is one main goal of the current paper, see for instance Corollary 2 and Corollary 3.

Lytchak and Petrunin [4] introduced the notion \(k\)-exceptional in their study of convex sets in generic manifolds. In particular, they introduced \(k\)-exceptional as a finite-order curvature obstruction for the existence of totally geodesic sub-manifolds.

The notation \(\mathfrak{P}\) is for parallel. The notion of \(\mathfrak{P}\)-spaces was introduced by Berndt and Vanhecke [18] to study generalizations of locally symmetric spaces, and to study classifications of Riemannian manifolds.
It is not difficult to see that all these three definitions are closely connected. In our main theorem (Theorem 3 below), we will make these connections more clear; roughly speaking, all these three notions of curvatures are equivalent.

2 Statement of main results↩︎

We calculate the matrix 4 for \(\dist_{\epsilon}(\bfx, y)\). More precisely, given a geodesic \(\gamma\), say the one given by 6 , if we replace \(\phi\) in 2 by the distance function \(\dist_{\epsilon}(\bfx, y)\), then \[X(t, 0; 0)=0, \;\forall t.\] With this preparation, one can compute the \(4\times\infty\) matrix 4 for the distance function \(\dist_{\epsilon}(\bfx, y)\) along the given geodesic \(\gamma\). For future reference, let us write this matrix as \[\label{260509e2952} \mathfrak{W}:= \left[\begin{array}{ccccc} \partial_s \mathcal{W}(0, \epsilon) & \partial_s^2 \mathcal{W}(0, \epsilon) & \dots & \partial_s^k \mathcal{W}(0, \epsilon) & \dots \\ \partial_s W_{11}(0, \epsilon) & \partial_s^2 W_{11}(0, \epsilon) & \dots & \partial_s^k W_{11}(0, \epsilon) & \dots \\ \partial_s W_{12}(0, \epsilon) & \partial_s^2 W_{12}(0, \epsilon) & \dots & \partial_s^k W_{12}(0, \epsilon) & \dots \\ \partial_s W_{22}(0, \epsilon) & \partial_s^2 W_{22}(0, \epsilon) & \dots & \partial_s^k W_{22}(0, \epsilon) & \dots \end{array}\right].\tag{10}\] We use \(\mathfrak{W}^-\) to denote the last three rows of \(\mathfrak{W}\) and \(\mathfrak{W}_k\) to denote the first \(k\) columns of the matrix \(\mathfrak{W}\). Moreover, we will recycle the setup in ?? and ?? .

Theorem 2 is relatively easy to prove. It reinforces the fact that the first row plays a particular role, as already discussed below Definition 3. As a consequence of Theorem 2, we know that for Riemannian distance functions, the matrix 10 satisfies the rank-\(r\) condition if and only if it has rank \(r\).

We summarize the above result in the following table. In the following table, we abbreviate “simultaneously diagonalizable” to (SD). Moreover, all the quantities \(R, R_{ij}\) and their derivatives are evaluated at \(s=0\).

\(\Leftrightarrow\)

The first item in Theorem 3, combined with the results in [2], has the following corollary.

The second item in Theorem 3 has the following corollary.

So far we have finished our discussion on rank-\(2\) conditions and rank-\((\le 3)\) conditions. Next, we will discuss rank-\(4\) condition.

That \((3.3)\) implies (3.1) when \(k=4\) has been obtained in [2]. ³ The proof there is via direct (which means brute-force) calculations, and we find it very complicated, if not impossible at all, to generalize to larger \(k\) and even to proving that (3.1) implies (3.3).

Some of the calculations in [2] will be reviewed when we prove Theorem 3, see for instance the beginning of Subsection 4.2. We will find that these calculations are still somewhat useful: They will help us accept certain patterns in the recursion relations for general \(k\), see for instance Proposition 11, and will also help us understand Theorem 3 better.

Let us be more precise here about what we mean by understanding Theorem 3 better. In [2], the authors first calculated the \(4\times \infty\) matrix 10 . In order to prove that (3.3) implies (3.1) when \(k=4\), the authors expanded the determinant of the leading \(4\times 4\) matrix in Taylor’s series in the \(\epsilon\) variable, and obtain that this determinant is equal to \[\label{260505e19524a} -\frac{1}{144}\pnorm{ (\mf{g}_{33, 11}-\mf{g}_{33, 22}) \mf{g}_{33, 123} - (\mf{g}_{33, 113}-\mf{g}_{33, 223}) \mf{g}_{33, 12} } \epsilon^5+ O(\epsilon^6),\tag{11}\] where \[\label{260505e19525a} R(0)= -\frac{1}{2} \begin{bmatrix} \mf{g}_{33, 11} & \mf{g}_{33, 12}\\ \mf{g}_{33, 12} & \mf{g}_{33, 22} \end{bmatrix}\tag{12}\] and \[\label{260505e19526a} R'(0)= -\frac{1}{2} \begin{bmatrix} \mf{g}_{33, 113} & \mf{g}_{33, 123}\\ \mf{g}_{33, 123} & \mf{g}_{33, 223} \end{bmatrix},\tag{13}\] see (3.109), (3.95) and (3.96) in [2]. Note that \(R(0)\) and \(R'(0)\) commuting means exactly that the coefficient of \(\epsilon^5\) in 11 vanishes, and this proves that (3.3) implies (3.1).

Let us see what (3.1) implying (3.3) means, which is equivalent to saying that if \(R(0)\) and \(R'(0)\) commute, then the determinant of the leading \(4\times 4\) matrix of 4 vanishes for every \(\epsilon>0\) (by analyticity), in other words, all the Taylor coefficients in the expansion 11 vanish. However, the calculation in 11 only tells us that the leading coefficient vanishes under the assumption that \(R(0)\) and \(R'(0)\) commute, and we still need to prove that \[(\mf{g}_{33, 11}-\mf{g}_{33, 22}) \mf{g}_{33, 123} - (\mf{g}_{33, 113}-\mf{g}_{33, 223}) \mf{g}_{33, 12}\] is a factor of every Taylor coefficient; this is what we proved in Theorem 3. In other words, if the coefficient of the lowest order monomial (which is \(\epsilon^5\) in the current case) vanishes, then all other coefficients will vanish as well.

This phenomenon still looks quite surprising to us, and it is not clear to us whether any similar phenomenon has been observed in the literature. This phenomenon seems to suggest that by only knowing some “finite order" curvature information, we may conclude curvature information for”infinite orders", which, combined with analyticity, will determine certain local geometry completely.
In the same paper [18] that Berndt and Vanhecke introduced \(\mathfrak{P}\)-spaces, they also introduced the notion of \(\mathfrak{C}\) spaces: Under the same setting as in Definition 6, we say that \(\mathcal{M}\) is a \(\mathfrak{C}\)-space if the eigenvalues of \(R(t)\) are constant in \(t\), for every geodesic \(\gamma\). Moreover, they proved that \[(\mathfrak{P}-\text{spaces})\cap (\mathfrak{C}-\text{spaces})= \text{locally symmetric spaces}.\] Let us bring the notion of \(\mathfrak{C}\) spaces into the whole picture of classifying curvature conditions described by Theorem 3.

Fix \(1\le r\le 4\) and \(k\in \N\). We say that a three dimensional analytic manifold \(\mathcal{M}\) is called an \((r, k)\)-manifold if \[\mathrm{rank}\; \mathfrak{W}= \mathrm{rank} \;\mathfrak{W}_k=r,\] for every geodesic \(\gamma\).
As an immediate corollary of Theorem 3 and the discussions around 5 , we obtain the following corollary.

Corollary 4 discusses Kakeya sets on manifolds satisfying the chaotic curvature condition of order \(\le (k-3)\), or equivalently, manifolds that are not \((k-1)\)-exceptional, for all \(k\ge 4\). However, it is not even clear whether such manifolds exist. The authors in [4] remarked that “probably every Riemannian metric is \(3\)-exceptional". The theorem below confirms that this is indeed the case.

Item (1) of Theorem 4 says that variably curved manifolds (defined below Definition 4) do not exist. A statement that is almost equivalent to Item (1) is that at every point on the manifold, we can find a geodesic \(\gamma\) passing through that point such that the manifold fails the chaotic curvature condition of order \(\le 1\) along \(\gamma\) at that point.

Of course one can easily find manifolds \(\mc{M}\) and geodesics \(\gamma\) on them such that \(\mc{M}\) satisfies the chaotic curvature condition of order \(\le 1\) at \(\gamma(0)\) along \(\gamma\), and therefore the more general version of Theorem 3.4 in [1], which discusses chaotic curvatures of order \(\le 1\) along a family of geodesics, is perfectly valid; indeed, this is the notion of chaotic curvature adopted by [16], where the authors proved that a “generic" three dimensional Riemannian manifold satisfies Sogge’s chaotic curvature condition of order \(\le 1\), under the extra assumption that we are also allowed to restrict the directions of geodesics when defining what”generic" means. ⁴
As a consequence of Theorem 4, Theorem 3 and the discussion around 5 , we obtain immediately the following corollary.

We also conjecture that for every three dimensional analytic manifold, there exists \(\kappa>0\) depending on the manifold such that 1 with \(R_{\varepsilon, N} f\) (see 7 for its definition) in place of \(T_N^{(\phi)}f\), holds for all \[p\ge \frac{10}{3}-\kappa.\] If this were true, it would imply that for every three dimensional analytic manifold, there exists \(\kappa'>0\) depending on the manifold such that its Kakeya sets have dimension at least \(2+\kappa'\).

Moreover, Minicozzi and Sogge [10] constructed a sequence of three dimensional analytic manifolds along which \(\kappa\to 0\). In other words, \(\kappa\) cannot be chosen to be a universal constant.

3 Proof of corollaries↩︎

The proofs for Corollary 2, Corollary 4 and Corollary 5 are immediate, and we will skip their proofs. In this section, we will only present the proofs of Corollary 1 and Corollary 3.

3.1 Proof of Corollary 1↩︎

Let us prove something more general.

Recall that it is proven in [2] that the distance function \(\dist_{\epsilon}(\bfx, y)\) satisfies Bourgain’s condition for every \(\bfx, y, \epsilon\) if and only if the manifold \(\mathcal{M}\) has constant sectional curvature. This, combined with Lemma 1, finishes the proof of Corollary 1.

3.2 Proof of Corollary 3↩︎

Let \(c_1\) be the trace of \(R(s)\), and \(c_2\) its determinant. Let us first prove that \[\label{260510e8951} c_2- \frac{c_1^2}{4}\le 0.\tag{16}\] Note that \[c_1= R_{11}+ R_{22}, \;\;c_2= R_{11}R_{22}- (R_{12})^2.\] We can compute \[\label{260510e8953} c_2- \frac{c_1^2}{4}= R_{11}R_{22}- (R_{12})^2- \frac{(R_{11}+ R_{22})^2}{4}= -(R_{12})^2- \frac{(R_{11}- R_{22})^2}{4}.\tag{17}\] This proves 16 .
Under the assumption that the eigenvalues of \(R(s)\) are constant in \(s\), we know that both \(c_1\) and \(c_2\) are constants in \(s\). Decompose \[R(s)= \frac{c_1}{2} I+ R_-(s),\] where \(R_-(s)\) is the trace-free part of \(R(s)\). One can compute directly that \[\det R_-(s)= c_2- \frac{c_1^2}{4}.\] Because of 16 , we denote \[c_2- \frac{c_1^2}{4}=-\mu^2\le 0,\] for some \(\mu\ge 0\).

Let us first prove Item (3). From 17 it is easy to see that \(\mu=0\) if and only if \[R_{11}(s)\equiv R_{22}(s), \;\;R_{12}(s)\equiv 0,\] which is the same as saying that \[R_-(s)=0, \;\forall s;\] in other words, the matrix \(R(s)\) is a constant scalar matrix.

Note that (3) has three statements: The first statement is \[\label{260517e89510} \text{the 4\times \infty matrix has rank two for all small \epsilon},\tag{18}\] the second statement is \[\label{260517e89511} \operatorname{dim}\left(\operatorname{span}\left\{I, R(0), R^{\prime}(0), R^{\prime \prime}(0), \ldots\right\}\right)=1,\tag{19}\] and the third statement is \[\label{260517e89512} R(s) \text{\;is constant in\;} s, \;\operatorname{dim}(\operatorname{span}\{I, R(0)\})=1 .\tag{20}\] In Theorem 3 we already obtained that the matrix 10 has rank two if and only if \(R(s)\) is simultaneously diagonalizable, and after diagonalizing we have \[R_{11}(s)=R_{22}(s), \;\forall s.\] Recall that currently we are assuming that all eigenvalues of \(R\) are constant. As a consequence, we obtain that the matrix 10 has rank two if and only if \(R(s)\) is a constant scalar matrix, that is, 18 is equivalent to 20 .

Note that 19 is the same as saying that \(R(s)\) is a scalar matrix, and therefore by the assumptions that all eigenvalues of \(R\) are constant, we can conclude that 19 is the same as saying that \(R(s)\) is a constant scalar matrix.

So far we have proven that all the three statements 18 , 19 and 20 are equivalent to that \(R(s)\) is a constant scalar matrix, and moreover they are all equivalent to that \(\mu=0\).
From now on let us assume that \(\mu>0\). We are about to prove (1) and (2). It is elementary to see that to prove the entire Corollary 3, we only need to prove one of the two statements (1) and (2). Here we will prove item (2).

Item (2) contains three statements: The first statement is \[\label{260517e89510z} \text{the 4\times \infty matrix has rank three for all small \epsilon},\tag{21}\] the second statement is \[\label{260517e89511z} \operatorname{dim}\left(\operatorname{span}\left\{I, R(0), R^{\prime}(0), R^{\prime \prime}(0), \ldots\right\}\right)=2,\tag{22}\] and the third statement is \[\label{260517e89512z} R(s) \text{\;is constant in\;} s, \;\operatorname{dim}(\operatorname{span}\{I, R(0)\})=2.\tag{23}\] That 21 is equivalent to 23 follows immediately from item (2) in Theorem 3. That 23 implies 22 is trivial. It remains to prove that 22 implies 23 .

Note that 22 is the same as saying that \[\label{260517e59522z} \operatorname{dim}\left(\operatorname{span}\left\{R_-(0), R_-^{\prime}(0), R_-^{\prime \prime}(0), \ldots\right\}\right)=1.\tag{24}\] By the fact that \(\mu>0\), we see that \(R_-(0)\) is not the zero matrix. This, combined with 24 , implies that \(R^{(m)}_-(0)\) is a multiple of \(R_-(0)\), for every \(m\ge 1\). We claim that \[R^{(m)}_-(0)=0, \;\forall m\ge 1.\] Once this is proven, we see immediately that 23 holds.

The proofs for these two claims are elementary, and we leave them out. By the Cayley-Hamilton theorem, we obtain \[\label{260509e8954} (R_-(s))^2= - (\det R_-(s)) I= \mu^2 I.\tag{25}\] We take the derivative in \(s\) on both sides of 25 and obtain \[\label{260509e8958} R_-(s) R'_-(s)+ R'_-(s) R_-(s)=0.\tag{26}\] Note that \(R_-(s)\) is trace-free, and therefore \(R'_-(s)\) is trace-free as well. We combine 26 , Claim 5 and Claim 6, and obtain that \(R'_-(s)\) is orthogonal to \(R_-(s)\) in \(\mathcal{S}_0\). However, previously we also obtained that \(R'_-(0)\) is a scalar multiple of \(R_-(0)\). All these imply that \[R'_-(0)=0.\] We can continue this process by differentiating more on both sides of 25 , and eventually obtain \[R^{(m)}_-(0)=0, \;\forall m\ge 1.\] This finishes the proof of the entire corollary.

4 Proof of Theorem 3: Item (3)↩︎

Item (3) consists of two parts. The first part is to show that the matrix 10 has rank four for all small enough \(\epsilon\) if and only if \(R(s)\) is not simultaneously diagonalizable in \(s\), and the second part is the equivalence of items (3.1)-(3.5). We will only prove the second part, and the first part follows immediately.
Let us be more precise. Assume that \(R(s)\) is not simultaneously diagonalizable in \(s\). We can therefore find \(k\) such that \(R(0), \dots, R^{(k-3)}(0)\) cannot be simultaneously diagonalized, that is, item (3.3) holds. By that (3.3) and (3.1) are equivalent, we can conclude that the matrix 10 has rank four for all small enough \(\epsilon\).

Now we assume that the matrix 10 has rank four for all small enough \(\epsilon\). We can conclude immediately that there exist \(k\) and \(\epsilon>0\) such that the first \(k\) columns of the matrix 10 has rank four. By that (3.2) and (3.3) are equivalent, we obtain that \(R(0), \dots, R^{(k-3)}(0)\) cannot be simultaneously diagonalized, and therefore \(R(s)\) is not simultaneously diagonalizable in \(s\).
In the rest of this section, we will prove the equivalence of (3.1)-(3.5). That (3.1) and (3.2) are equivalent follows immediately from analyticity. In Subsection 4.2 we will prove that (3.2) implies (3.3), in Subsection 4.3 we will prove that (3.3) implies (3.2), and in the last subsection we will prove that (3.3), (3.4) and (3.5) are mutually equivalent.

4.1 Preliminary calculations↩︎

Let \(\gamma\) be a geodesic with unit speed. We make the same normalization as in 6 that \[\label{260504e3957} \gamma(0)=(0, 0, 0), \;\;\gamma(s)=(0, 0, s), \forall s.\tag{27}\] We take the phase function \[\label{260504e3958} \phi_{\epsilon}(x, t; \xi):= \dist((x, t), (\xi, \epsilon)),\tag{28}\] and compute the \(4\times \infty\) matrix 4 at \[\label{260504e3959} x_0=(0, 0), \;\;\xi=(0, 0).\tag{29}\] Characteristic curves for the distance function are precisely given by geodesics (see for instance Lemma 3.2 in [2]), and therefore \[X(t; x_0; \xi)=0, \;\;\forall t,\] due to the normalization in 27 , where \(X\) was introduced in 2 .

We will rely heavily on several calculations from [2]. In order for our notation to be consistent with the ones in [2], we will switch from \(\xi\) to \(y\). We follow (3.32)–(3.34) in [2], and denote \[\label{260504e39512} \Phi(x, t; y, \tau):= \dist((x, t), (y, \tau)),\tag{30}\] \[\phi_{0, \epsilon}(x, t ; y):=\phi_\epsilon(x, t ; y)-\phi_\epsilon(0,0 ; y),\] and \[\label{260504e39514} \Phi_0(x, t ; y, \tau):=\Phi(x, t ; y, \tau)-\Phi(0,0 ; y, \tau) .\tag{31}\] Note that in 30 and 31 , we are no longer fixing \(\tau=\epsilon\) but treating \(\tau\) as a variable; this is necessary as we will be computing covariant derivatives of the distance function in all its variables (not just \(y\)).

Recall from 3 that we need to compute \[\partial_{y_i} \partial_{y_j} \phi_{0, \epsilon}.\] By [2], we have \[\label{260601e5959zz} \left.\frac{\partial}{\partial y_i} \frac{\partial}{\partial y_j} \phi_{0, \epsilon}\right|_{\substack{(x, t)=\gamma(s)\\ y=0}}= \left.\left(\operatorname{Hessian} \Phi_0\right)\right|_{\substack{(x, t)=\gamma(s)\\(y, \tau)=(0, \epsilon)}}\left(\frac{\partial}{\partial y_i}, \frac{\partial}{\partial y_j}\right)\tag{32}\] for every \(s\in [0, \epsilon)\) and \(1\le i, j\le 2\), where \(\mathrm{Hessian}\) refers to the covariant Hessian in the \((y, \tau)\) variables.
Let us record here an equation that is slightly stronger than 32 .

Back to 32 . We follow (3.63)-(3.66) in [2] and denote \[W_{ij}(s, \epsilon):= \left.\left(\operatorname{Hessian} \Phi_0\right)\right|_{\substack{(x, t)=\gamma(s)\\(y, \tau)=(0, \epsilon)}}\left(\frac{\partial}{\partial y_i}, \frac{\partial}{\partial y_j}\right),\] \[W(s, \epsilon):=\left[\begin{array}{ll} W_{11}(s, \epsilon) & W_{12}(s, \epsilon) \\ W_{21}(s, \epsilon) & W_{22}(s, \epsilon) \end{array}\right],\] and \[\mathcal{W}(s, \epsilon):= \det W(s, \epsilon).\] We will need to compute the rank of the matrix \[\label{260504e39520} \left[\begin{array}{ccccc} \partial_s \mathcal{W}(0, \epsilon) & \partial_s^2 \mathcal{W}(0, \epsilon) & \dots & \partial_s^k \mathcal{W}(0, \epsilon) & \dots \\ \partial_s W_{11}(0, \epsilon) & \partial_s^2 W_{11}(0, \epsilon) & \dots & \partial_s^k W_{11}(0, \epsilon) & \dots \\ \partial_s W_{12}(0, \epsilon) & \partial_s^2 W_{12}(0, \epsilon) & \dots & \partial_s^k W_{12}(0, \epsilon) & \dots \\ \partial_s W_{22}(0, \epsilon) & \partial_s^2 W_{22}(0, \epsilon) & \dots & \partial_s^k W_{22}(0, \epsilon) & \dots \end{array}\right]\tag{33}\] for every \(\epsilon\). We repeat (3.67)–(3.88) in [2], and obtain that \[\label{260505e39521} W_{ij}(s, \epsilon)= \partial_{s'} a_{ij}(s, \epsilon)-\partial_{s'} a_{ij}(0, \epsilon),\tag{34}\] where \[A(s, s')=\begin{bmatrix} a_{11}(s, s') & a_{12}(s, s')\\ a_{21}(s, s') & a_{22}(s, s') \end{bmatrix}\] solves the system of ODEs \[\label{260504e39523} \begin{align} & \partial_{s^{\prime}}^2 A\left(s, s^{\prime}\right)+A\left(s, s^{\prime}\right) R\left(s^{\prime}\right)=0 \\ & A(s, s)=0, A(s, \epsilon)=I_{2 \times 2} \end{align}\tag{35}\] with \(R(s')\) given by ?? and ?? .

4.2 Item (3.2) implies item (3.3)↩︎

We will argue by contradiction, and assume that \(R(0), R'(0), \dots, R^{k-3}(0)\) can be simultaneously diagonalized. Our goal is to show that for every \(\epsilon>0\), the first \(k\) columns of the matrix 10 has rank \(\le 3\).
We keep the normalization in 27 , and recycle the notation in 28 – 33 . To show the matrix 33 has rank \(\le 3\), we will need to solve the ODE system 35 .

Let us start by recalling some more⁶ preliminary calculations in [2], and we will also take this opportunity to raise a question about a potentially more quantitative version of item (3.1) in the statement of Theorem 3.

In [2], in order to solve the ODE system 35 , two auxiliary ODE systems were introduced. They are \[\left\{ \begin{align} &\partial_t^2 B_1(t)+B_1(t)R(t)=0,\\ &B_1(0)=I,\qquad \partial_t B(0)=0, \end{align} \right.\] and \[\left\{ \begin{align} &\partial_t^2 B_2(t)+B_2(t)R(t)=0,\\ &B_2(0)=0,\qquad \partial_t B(0)=I, \end{align} \right.\] see (3.113) and (3.114) in [2]. We can then write \[A(s, t)= C_1(s) B_1(t)+ C_2(s) B_2(t),\] for some scalar matrices \(C_1(s), C_2(s)\), and we can use the boundary conditions on \(A(s, t)\) to determine the derivatives of \(C_1, C_2\) at \(s=0\).

For the three equations in Lemma 2, we refer to (3.123), (3.125), (3.129) in [2]. After obtaining Lemma 2, the authors of [2] expanded \(\partial^{k'}_s W(0, \epsilon)\), which is equal to \[\partial^{k'}_s \partial_{s'} A(0, \epsilon)\] by 34 , in Taylor’s series in the \(\epsilon\) variable, for all \(k'=1, 2, 3, 4\).

In the end, [2] computes the determinant of the leading \(4\times 4\) matrix of 33 directly, and obtain that it is given by \[\label{260505e4956} -\frac{1}{144}\pnorm{ (\mf{g}_{33, 11}-\mf{g}_{33, 22}) \mf{g}_{33, 123} - (\mf{g}_{33, 113}-\mf{g}_{33, 223}) \mf{g}_{33, 12} } \epsilon^5+ O(\epsilon^6),\tag{36}\] see 12 and 13 for the new notation above.
Note that if \(R(0), R'(0)\) do not commute, that is item (3.3) of Theorem 3 holds with \(k=4\), then the coefficient of \(\epsilon^5\) in 36 does not vanish, and therefore the first four columns of the matrix 33 has rank four, whenever \(\epsilon\) is chosen to be small enough. This is exactly how [2] proves that when \(k=4\), item (3.3) of Theorem 3 implies item (3.1).

With the help of the calculation 36 and with the assumption \(k=4\), we actually know that whenever item (3.1) of Theorem 3 holds, it always holds very “quantitatively", that is, as \(\epsilon\to 0\), the determinant cannot tend to zero faster than \(\epsilon^5\).

When trying to prove Theorem 3 for general \(k\ge 4\), we tried to generalize the above approach, that is, the approach of Taylor’s expansions in \(\epsilon\), but failed dramatically. We even failed dramatically at \(k=4\) when computing the coefficient of \(\epsilon^{j}\) in 36 for large \(j\in \N\).

As a consequence, when we are in the case that item (3.1) of Theorem 3 holds, we only know that there exists one \(4\times 4\) minor coming from the first \(k\) columns whose determinant is not identically zero in \(\epsilon\), but we do not know the non-vanishing order of this determinant (other than the case \(k=4\) which was calculated in 36 ).
Before we start explaining our new approach, let us record certain simplifications of the expressions in Lemma 2. The idea behind this simplifying process will also be used in our new approach. Moreover, we will see that the simplified expressions will help us understand the recursion relations for general \(k\), see for instance Proposition 11.

Let us begin explaining our new approach. Consider the initial value problem: \[\label{260504e4951a} \left\{ \begin{align} &\partial_t^2 B(s,t)+B(s,t)R(t)=0,\\ &B(s,s)=0,\qquad \partial_t B(s,s)=I. \end{align} \right.\tag{40}\]

Note that \[\label{260602e49537} W(s,\epsilon)=\sec(s,\epsilon)-\sec(0,\epsilon).\tag{42}\] To study the second fundamental form \(\sec(s, \epsilon)\), we introduce the following two crucial ODE systems: \[\left\{ \begin{align} &A_1''(t)+A_1(t)R(t)=0,\\ &A_1(\epsilon)=I,\qquad A_1'(\epsilon)=0,\\ \end{align} \right.\] and \[\label{260505e49517tt} \left\{ \begin{align} &A_2''(t)+A_2(t)R(t)=0,\\ &A_2(\epsilon)=0,\qquad A_2'(\epsilon)=-I. \end{align} \right.\tag{43}\]

We find an important relation between \(\sec(s,\epsilon)\) and \(\sec(\epsilon,s)\):

One first difficulty in computing the \(4\times \infty\) matrix 33 is how to calculate partial derivatives in the \(s\) variable. More precisely, from the defining equation 40 , we already know how to take successive derivatives in the \(t\) variable, which is by taking derivatives in \(t\) on both sides of the first equation in 40 . However, finding a convenient way of taking derivatives in the \(s\) variable is more challenging.

The dual relation ?? in Proposition 9 and the reversed Riccati equation ?? tell us exactly how to take all derivatives in the \(s\) variable.

This relation between \(\sec(s, \epsilon)\) and \(\sec(\epsilon, s)\) seems to be new in the literature, although equations in the same spirit have been found in [19]: \[\begin{align} \sec(s,\epsilon)&=A_1(s)A_2^{-1}(s),\tag{47}\\ \partial_s\sec(s,\epsilon)&=(A_2^T)^{-1}(s)A_2^{-1}(s),\tag{48} \end{align}\] that is, the authors in [19] only need to take derivatives up to order one in their problem. However, in the current problem, we cannot reveal the relation between \(\sec(s,\epsilon)\) and \(\sec(\epsilon,s)\) without taking the second derivative. Moreover, we also obtain the recursion relation ?? of all orders. To be self-contained, we will provide the proofs of 47 and 48 .

Next, we will do a decomposition for \(I_k(\epsilon,s)\) and find a recursive relation.

Let us record one more recursive relation that will be very useful later. Denote \[\mathcal{P}_k(s):= \mathcal{P}_k(R(s), \dots, R^{(k-3)}(s)).\] From 57 and ?? , we see that \[\label{260506e49594} \mathcal{P}_{k+1}(s)= \partial_s \mathcal{P}_k(s)+ (\mathrm{tr}(\sec(\epsilon, s))) \mathcal{P}_k(s)+ a_k(s)R(s),\tag{58}\] where \(a_k\) is the same as the one in ?? . One can compute directly (see for ?? ) that \[\mathcal{P}_1(s)=\mathcal{P}_2(s)=0.\] From this we can compute that \[\mathcal{P}_3(s)= a_2(s) R(s).\] From Proposition 11, we obtain that \(a_2(s)=2\), and therefore \[\mathcal{P}_3(s)=2 R(s).\] We apply 58 iteratively, and obtain that \(\mathcal{P}_i(0)\) is a linear combination of \(R(0), R'(0), \dots\), and that \[\label{260516e69573} \mathcal{P}_m(0)= 2 R^{(m-3)}(0)+ \text{lower derivatives of } R,\tag{59}\] for all \(m\ge 3\).
Next, we will compute the \(\partial_s^k\det W(s,\epsilon)\) where \[W(s,\epsilon)=\sec(s,\epsilon)-\sec(0,\epsilon).\] For a \(2\times 2\) matrix \[S=\left(\begin{array}{cc} a & b \\ b & c \end{array}\right)\] we write the companion of \(S\) as \[S^{\sharp}=\left(\begin{array}{cc} c & -b \\ -b & a \end{array}\right)\] Then we have the following Jacobi’s formula \[\label{260505e49536} \frac{d}{ds}\det S(s)=\mathrm{tr} (S^{\sharp}S'),\tag{60}\] where \(S^{\sharp}\) is the companion of \(S\). More importantly, we will see that coefficients of \(\partial_s^k\det W(0,\epsilon)\) satisfy the same recursive relation with the coefficients \(a_k\) in the decomposition ?? of \(I_k\).

Proposition 13 leads to a contradiction that we are assuming item (3.2) in Theorem 3. This finishes the proof that (3.2) implies (3.3).

4.3 Item (3.3) implies item (3.2)↩︎

Let \(k \ge 4\). We argue by contradiction, and assume that for every \(\epsilon\) small enough, the first \(k\) columns of the matrix 4 has rank \(\le 3\). Our goal is to derive a contradiction to item (3.3).
We continue with the setup in the proof that (3.2) implies (3.3). In particular, we will consider the rank of the matrix ?? . Fix an \(\epsilon>0\) small enough. There exist \(c_i=c_i(\epsilon)\in \R, i=1, \dots, 4\), not all zero, such that \[\begin{gather} \label{260506e49572} c_1(\epsilon) \partial_s^i W_{11}(0, \epsilon) + c_2(\epsilon) \partial_s^i W_{12}(0, \epsilon) \\ + c_3(\epsilon) \partial_s^i W_{22}(0, \epsilon) + c_4(\epsilon) \partial_s^i \det W(0, \epsilon) = 0, \;\forall i=1, 2, \dots, k. \end{gather}\tag{62}\] Let us write the above relation in a more compact form: Denote \[K= \begin{bmatrix} c_1 & c_2/2\\ c_2/2 & c_3 \end{bmatrix}\] and we can write 62 equivalently as \[\label{260506e49575} \operatorname{tr}\big(K(\epsilon) \partial_s^i W(0, \epsilon)\big) + c_4 \partial_s^i \det W(0, \epsilon) = 0.\tag{63}\] Recall from ?? that \[\partial^i_s W(0, \epsilon)= (A_2^T(0))^{-1} I_i(\epsilon, 0) (A_2(0))^{-1},\] and from Proposition 12 that \[\partial^i_s \det W(0, \epsilon)= a_i(0) (\det A_2(0))^{-2}= d_i(0) (\det A_2(0))^{-2}\] where we applied 61 and \[W^{\sharp}(0, \epsilon)=0,\] we can write 62 further as \[\label{260506e49579} \operatorname{tr}\big(K (A_2^T)^{-1} I_i(\epsilon, 0) (A_2)^{-1}\big) + c_4 a_i(0) \det(A_2)^{-2} = 0.\tag{64}\] By using elementary properties of traces, the first term can be written as \[\operatorname{tr}\big( (A_2)^{-1} K (A_2^T)^{-1} I_i(\epsilon, 0) \big).\] Let us denote \[Y= (A_2)^{-1} K(A_2^T)^{-1},\] and \[\lambda= c_4 \det(A_2)^{-2}.\] Under these notations, 64 is equivalent to \[\label{260506e49583} \mathrm{tr} (Y(\epsilon) I_i(\epsilon, 0))+ \lambda(\epsilon) a_i(0)=0, \;i=1, \dots, k.\tag{65}\] Note that \[(c_1, c_2, c_3, c_4) \neq 0\] guarantees that \[\label{260506e49585} (Y, \lambda) \neq (0, 0).\tag{66}\] Now we consider some special values of \(i\). First of all, we take \(i=1\). We have that \(I_1=I\), the \(2\times 2\) identity matrix, and \(a_1(0)=0\). As a consequence, we obtain that \[\mathrm{tr}(Y(\epsilon))=0,\] that is, \(Y\) is a trace-free symmetric matrix. Next, we take \(i=2\). In this case, from Proposition 9, we obtain that \[\label{260506e49587} I_2(\epsilon, 0)= 2\sec(\epsilon, 0),\tag{67}\] and from Proposition 11, we obtain that \(a_2(0)=2.\) By substituting these into 65 , we obtain \[2\mathrm{tr}(Y(\epsilon) \sec(\epsilon, 0))+ 2 \lambda(\epsilon) =0,\] which implies that \[\label{260506e49589} \lambda(\epsilon)= -\mathrm{tr}(Y(\epsilon) \sec(\epsilon, 0)).\tag{68}\] From 68 we can also conclude that \(Y\neq 0\), as otherwise both \(Y\) and \(\lambda\) need to vanish, which contradicts 66 .
Now we go back to 65 and consider what it says for a more general \(i\ge 3\). We first follow ?? , and write \[\label{260506e49590} I_i(\epsilon, 0)= a_i(0) \sec(\epsilon, 0) + b_i(0) I+ \mathcal{P}_i(R(0), \dots, R^{(i-3)}(0)).\tag{69}\] We substitute this into 65 with \(i\ge 3\), and obtain \[a_i(0) \mathrm{tr}(Y(\epsilon) \sec(\epsilon, 0)) + \mathrm{tr}(Y(\epsilon) \mathcal{P}_i)+ a_i(0) \lambda(\epsilon)=0,\] where we used the fact that \(Y\) is trace-free. This, combined with 68 , implies that \[\label{260506e49592} \mathrm{tr}(Y(\epsilon) \mathcal{P}_i)=0, \;k\ge i\ge 3.\tag{70}\] The recursive relation 59 , combined with 70 , implies that \[\mathrm{tr}(Y(\epsilon) R^{(i)}(0))=0, \;i=0, 1, \dots, k-3.\] We claim that \[\label{260506e495100} \mathrm{span}\{I, R(0), \dots, R^{(k-3)}(0)\}\le 2.\tag{71}\] To see 71 , we just need to write \[R^{(i)}(0)= R^{(i)}(0)- \frac{\mathrm{tr} R^{(i)}(0)}{2} I+ \frac{\mathrm{tr} R^{(i)}(0)}{2} I.\] By the fact that \(Y(\epsilon)\) is trace-free, we obtain that the trace of \[Y(\epsilon)\pnorm{ R^{(i)}(0)- \frac{\mathrm{tr} R^{(i)}(0)}{2} I }\] also vanishes. From this and the fact that \(Y\neq 0\), we can conclude 71 immediately.
Once 71 is proven, we see immediately that \(R(0), \dots, R^{(k-3)}(0)\) are simultaneously diagonalizable, and therefore we arrive at a contradiction.

4.4 The equivalence of (3.3), (3.4) and (3.5)↩︎

We now prove that Items (3.3), (3.4) and (3.5) in Theorem 3 are equivalent. The proof is elementary, but we give the details because it is important to keep track of the different notations for the Jacobi operators.
Let \(\mathcal{R}\) be the Riemannian curvature tensor. Set \[Z:=\gamma'(0), \qquad \bfP:=Z^\perp\subset T_{\gamma(0)}\mathcal{M}.\] For \(0\leq m\leq k-3\), define the linear operator \(\mathcal{R}_m:T_{\gamma(0)}M\to T_{\gamma(0)}M\) by \[\mathcal{R}_m(v):=(\nabla_Z^m \mathcal{R})(v,Z)Z.\] Thus, in the notation of Definition 5, \[\mathcal{R}_m=\mathcal{R}^{m+2}_{\gamma(0)}.\] Moreover, \[\mathcal{R}_m(Z)=0.\] The restriction of \(\mathcal{R}_m\) to \(\bfP\) is self-adjoint. If \(E_1(t),E_2(t)\) is the parallel orthonormal frame used in Definition 6, then the matrix of \(\mathcal{R}_m|_{\bfP}\) in the basis \(E_1(0),E_2(0)\) is exactly \[R^{(m)}(0),\] where the matrix \(R\) is defined in ?? . Indeed, \[\begin{gather} \left\langle \mathcal{R}_m(E_i(0)),E_j(0)\right\rangle= \left\langle (\nabla_{\gamma'}^m \mathcal{R})(E_i,\gamma')\gamma', E_j \right\rangle\big|_{s=0} \\ = \frac{d^m}{ds^m} \left\langle \mathcal{R}(E_i(s),\gamma'(s))\gamma'(s),E_j(s)\right\rangle \bigg|_{s=0} = R_{ij}^{(m)}(0). \end{gather}\]

We shall use the following elementary linear algebra fact. Let \(A_0,\ldots,A_N\) be self-adjoint operators on a two-dimensional real inner product space \(\bfP\). Then the following two statements are equivalent:

\[A_0,\ldots,A_N \quad \text{are simultaneously diagonalizable;}\] and there exists an orthonormal basis \((X,E)\) of \(\bfP\) such that \[\langle A_mX,E\rangle=0, \qquad 0\leq m\leq N.\] We first compare Item (3.3) with chaotic curvature. Let \(X=X(0)\in \bfP\) be a unit vector, and let \(X(t)\) be its parallel transport along \(\gamma\). Let \(E=E(0)\in \bfP\) be a unit vector perpendicular to \(X\), and let \(E(t)\) be its parallel transport along \(\gamma\). Then \[(X(t),E(t),\gamma'(t))\] is an orthonormal frame along \(\gamma\). By Definition 4, \[Y(t)=\bar{\operatorname{Ric}}(X(t)).\] Since \(E(t)\) spans the orthogonal complement of \(\operatorname{span}\{X(t),\gamma'(t)\}\), we have \[Y^\perp(t) = \operatorname{Ric}_{\gamma(t)}(X(t),E(t))E(t).\] Using the curvature symmetries in dimension three, and the fact that \(X(t),E(t)\) and \(\gamma'(t)\) are parallel along \(\gamma\), we obtain, for \(0\leq m\leq k-3\), \[\begin{gather} \nabla_{\gamma'}^mY^\perp(0) = (\nabla_{\gamma'}^m\operatorname{Ric})(X,E)E \\ = \left\langle (\nabla_Z^m \mathcal{R})(X,Z)Z,E \right\rangle E = \left\langle \mathcal{R}_m(X),E \right\rangle E. \end{gather}\] Therefore \[|Y^\perp(0)|+\cdots+ |\nabla_{\gamma'}^{k-3}Y^\perp(0)|\neq 0\] is equivalent to saying that \[\left\langle \mathcal{R}_m(X),E \right\rangle\neq 0\] for at least one \(0\leq m\leq k-3\). This, combined with the linear algebra observation above, proves that (3.3) and (3.4) are equivalent.
It remains to compare Item (3.3) with Item (3.5). By Definition 5, the geodesic \(\gamma\) is \((k-1)\)-exceptional if and only if there exists a two-dimensional subspace \[V\subset T_{\gamma(0)}\mathcal{M}, \qquad Z\in V,\] such that \[\mathcal{R}^q_{\gamma(0)}(V)\subset V, \qquad 2\leq q\leq k-1.\] Writing \(q=m+2\), this is the same as \[\mathcal{R}_m(V)\subset V, \qquad 0\leq m\leq k-3.\] Since \(V\) is two-dimensional and contains \(Z\), there exists a unit vector \(X\in \bfP\) such that \[V=\operatorname{span}\{Z,X\}.\] Let \(E\in \bfP\) be a unit vector perpendicular to \(X\). Since \(\mathcal{R}_m(Z)=0\) and \(\mathcal{R}_m(X)\in \bfP\), the condition \(\mathcal{R}_m(V)\subset V\) is equivalent to \[\mathcal{R}_m(X)\in \mathbb{R}X,\] or equivalently, \[\left\langle \mathcal{R}_m(X),E \right\rangle=0, \qquad 0\leq m\leq k-3.\] By the same linear algebra observation, this finishes the proof of the equivalence between (3.3) and (3.5).

5 Proof of Theorem 3: Item (2)↩︎

Item (2) follows immediately from the following three lemmas.

The proof of Lemma 8 is almost identical to that of Proposition 13, and we will not repeat it here. We will only present the proofs of Lemma 6 and Lemma 7.
Let us first prove Lemma 6. We argue by contradiction. If \(R(s)\) is not simultaneously diagonalizable, then we know from item (3) of Theorem 3 that the matrix 10 has rank four for sufficiently small \(\epsilon\), which is a contradiction. We therefore from now on assume that \(R(s)\) is simultaneously diagonalizable. If we further assume that \(R_{11}\equiv R_{22}\), then it is elementary to see that the matrix 10 has rank two, which is a contradiction. This finishes the proof of Lemma 6.
Let us prove Lemma 7. We argue by contradiction, and assume that for some \(\epsilon\) small enough, the matrix ?? has rank \(\le 2\). Our goal is to prove that \[R_{11}^{(m)}(0)=R_{22}^{(m)}(0), \;\forall 0\le m\le k-3.\] By the assumption, we can find \(c_1(\epsilon), c_2(\epsilon), c_3(\epsilon)\), not all zero, such that \[\label{260507e5952} c_1 \partial_s^i W_{11}(0, \epsilon) + c_2 \partial_s^i W_{22}(0, \epsilon) + c_3 \partial_s^i \det W(0, \epsilon) = 0.\tag{72}\] Let \(K=K(\epsilon)=\mathrm{diag}(c_1, c_2)\). Then 72 can be written equivalently as \[\operatorname{tr}\big(K \partial_s^i W(0, \epsilon)\big) + c_3 \partial_s^i \det W(0, \epsilon) = 0.\] We repeat the previous proof (around 65 ) and obtain \[\label{260506e49583zz} \mathrm{tr} (Y(\epsilon) I_i(\epsilon, 0))+ \lambda(\epsilon) a_i(0)=0, \;i=1, \dots, k.\tag{73}\] Note that \(Y(\epsilon)\) is not only trace-free as before but also a diagonal matrix. This, combined with \[\mathrm{tr}(Y(\epsilon) R^{(i)}(0))=0, \;i=0, 1, \dots, k-3,\] implies the desired result. This finishes the proof of Lemma 7.
Let us also remark here that after proving item (2) and item (3) in Theorem 3, item (1) holds automatically.

6 Proof of Theorem 2↩︎

Recall the setup in Subsection 4.1. For clarity in this proof, write \[\mathcal{W}(s,\epsilon):=\det W(s,\epsilon).\] Thus the first row of the matrix 10 consists of the derivatives of \(\mathcal{W}\), while the last three rows consist of the derivatives of \(W_{11},W_{12},W_{22}\).

Fix \(\epsilon>0\) sufficiently small. We argue by contradiction. Suppose that the first row is a linear combination of the second, third and fourth rows. Then there exist constants \(a,b,c\), depending possibly on \(\epsilon\), such that \[\partial_s^j\mathcal{W}(0,\epsilon) = a\,\partial_s^jW_{11}(0,\epsilon) + 2b\,\partial_s^jW_{12}(0,\epsilon) + c\,\partial_s^jW_{22}(0,\epsilon)\] for every \(j\geq 1\). Since \[W(0,\epsilon)=0, \qquad \mathcal{W}(0,\epsilon)=0,\] the same identity also holds for \(j=0\). By analyticity in the \(s\)-variable, we obtain \[\mathcal{W}(s,\epsilon) = aW_{11}(s,\epsilon)+2bW_{12}(s,\epsilon)+cW_{22}(s,\epsilon)\] for every \(0\leq s<\epsilon\).
Now set \[D:= \begin{pmatrix} c & -b\\ -b & a \end{pmatrix}.\] Since \(W(s,\epsilon)\) is symmetric, we may write \[W(s,\epsilon)= \begin{pmatrix} W_{11}(s,\epsilon) & W_{12}(s,\epsilon)\\ W_{12}(s,\epsilon) & W_{22}(s,\epsilon) \end{pmatrix}.\] A direct calculation gives \[\begin{gather} \det\bigl(W(s,\epsilon)-D\bigr) = \det W(s,\epsilon) - aW_{11}(s,\epsilon) \\ - 2bW_{12}(s,\epsilon) - cW_{22}(s,\epsilon) + \det D. \end{gather}\] Using the preceding linear identity, this becomes \[\det\bigl(W(s,\epsilon)-D\bigr)=\det D\] for every \(0\leq s<\epsilon\). Multiplying by \((\epsilon-s)^2\), we get \[\label{260603e6959} \det\bigl((\epsilon-s)W(s,\epsilon)-(\epsilon-s)D\bigr) = (\epsilon-s)^2\det D.\tag{74}\] It remains to understand the behavior of \(W(s,\epsilon)\) as \(s\to\epsilon^-\). We record the required form of Ledger’s recursion formula. Put \[r:=\epsilon-s\] and, for fixed \(s\), define \[S_s(r):=\sec(s,s+r),\] the second fundamental form that was used in Lemma 5.

Write the Taylor expansion of \(R\) at \(s\) as \[R(s+r)=\sum_{m=0}^{\infty}\frac{r^m}{m!}R^{(m)}(s).\] Ledger’s expansion has the form \[S_s(r)=\frac{1}{r}I+\sum_{k=0}^{\infty}r^k S_k(s).\] Substituting this expansion into the Riccati equation and comparing powers of \(r\), we obtain first \[S_0(s)=0,\] and then, for every \(k\geq 0\), \[S_{k+1}(s) = -\frac{1}{k+3} \left( \frac{R^{(k)}(s)}{k!} + \sum_{\ell=0}^{k}S_\ell(s)S_{k-\ell}(s) \right).\] In particular, \[S_1(s)=-\frac{1}{3}R(s).\] Therefore, as \(r=\epsilon-s\to0^+\), \[\sec(s,\epsilon) = \frac{1}{\epsilon-s}I - \frac{\epsilon-s}{3}R(s) + O\bigl((\epsilon-s)^2\bigr).\] Since \(\sec(0,\epsilon)\) is fixed as \(s\to\epsilon^-\), and since \[W(s,\epsilon)=\sec(s,\epsilon)-\sec(0,\epsilon),\] we obtain \[(\epsilon-s)W(s,\epsilon) = I-(\epsilon-s)\sec(0,\epsilon)+O\bigl((\epsilon-s)^2\bigr).\] Consequently, \[\lim_{s\to\epsilon^-}(\epsilon-s)W(s,\epsilon)=I.\] It follows that \[\lim_{s\to\epsilon^-} \det\bigl((\epsilon-s)W(s,\epsilon)-(\epsilon-s)D\bigr) = \det I = 1.\] On the other hand, from the identity 74 , we obtain \[\lim_{s\to\epsilon^-} \det\bigl((\epsilon-s)W(s,\epsilon)-(\epsilon-s)D\bigr) = 0.\] This contradiction proves that the first row of 10 is not a linear combination of the second, third and fourth rows.

7 Proof of Theorem 4↩︎

7.1 Proof of the first item↩︎

We first prove a purely algebraic lemma.

We now prove Theorem 4, Item (1). We prove the slightly stronger pointwise statement that, for every \(p\in \mathcal{M}\), there exists a unit speed geodesic \(\gamma\) with \(\gamma(0)=p\) and a unit vector \(X(0)\in T_p \mathcal{M}\), \(X(0)\perp \dot{\gamma}(0)\), such that \[Y^\perp(0)=0,\qquad \nabla_{\dot{\gamma}}Y^\perp(0)=0,\] where \(Y(t)=\bar{\operatorname{Ric}}(X(t))\) and \(Y^\perp(t)\) is as in Definition 4. This shows that the chaotic curvature condition of order \(\leq 1\) fails at \(p\) along the geodesic \(\gamma\).

Fix \(p\in \mathcal{M}\), and apply Lemma 9 to \(V=T_p \mathcal{M}\) with \[A(U,V):=\operatorname{Ric}_p(U,V), \qquad B(Z,U,V):=(\nabla_Z\operatorname{Ric})_p(U,V).\] The tensor \(A\) is symmetric, and \(B\) is symmetric in its last two variables because the Ricci tensor is symmetric. Hence there exists an orthonormal frame \((X_0,E_0,Z_0)\) of \(T_p \mathcal{M}\) such that \[\operatorname{Ric}_p(X_0,E_0)=0, \qquad (\nabla_{Z_0}\operatorname{Ric})_p(X_0,E_0)=0.\]

Let \(\gamma\) be the unit speed geodesic with \[\gamma(0)=p,\qquad \dot{\gamma}(0)=Z_0.\] Let \(X(t)\) and \(E(t)\) be the parallel transports of \(X_0\) and \(E_0\) along \(\gamma\), respectively. Thus \[X(0)=X_0,\qquad E(0)=E_0,\qquad \nabla_{\dot{\gamma}}X=\nabla_{\dot{\gamma}}E=0.\] By Definition 4, \[Y(t)=\bar{\operatorname{Ric}}(X(t)).\] Since the orthogonal complement of \(\operatorname{span}\{X(t),\dot{\gamma}(t)\}\) is spanned by \(E(t)\), we have \[\begin{align} Y^\perp(t) &= \big\langle \bar{\operatorname{Ric}}(X(t)),E(t)\big\rangle E(t) \\ &= \operatorname{Ric}_{\gamma(t)}(X(t),E(t))E(t). \end{align}\] Therefore \[Y^\perp(0)=\operatorname{Ric}_p(X_0,E_0)E_0=0.\] Moreover, since \(X(t)\) and \(E(t)\) are parallel along \(\gamma\), we have \[\nabla_{\dot{\gamma}}Y^\perp(t) = (\nabla_{\dot{\gamma}(t)}\operatorname{Ric})(X(t),E(t))E(t).\] Evaluating at \(t=0\), we get \[\nabla_{\dot{\gamma}}Y^\perp(0) = (\nabla_{Z_0}\operatorname{Ric})_p(X_0,E_0)E_0 = 0.\] Thus \[|Y^\perp(0)|+\left|\nabla_{\dot{\gamma}}Y^\perp(0)\right|=0,\] so the chaotic curvature condition of order \(\leq 1\) fails at the point \(p\) along the geodesic \(\gamma\). Since \(p\in \mathcal{M}\) was arbitrary, no three-dimensional Riemannian manifold satisfies the chaotic curvature condition of order \(\leq 1\). This proves Item (1) of Theorem 4.

7.2 Proof of the second item↩︎

We mainly follow the notation from [2], in particular, the notation in [2]. We will first find a metric satisfying the chaotic curvature condition of order \(\le 2\); by compactness and continuity, this immediately implies that every small smooth perturbation of it also satisfies chaotic curvature condition of order \(\le 2\). Next we will find another metric which fails the chaotic curvature condition of \(\le 2\), and so does every small smooth perturbation of it.
Let us work in a small neighborhood of \(0\in \R^3\), with coordinates \[(x, y, z)=(x^1, x^2, x^3).\] Let \(g\) be a given metric. We use \[e_i= \partial x^i\] to denote one vector in the standard basis for the tangent space. Denote \[g_{ij}:= g(e_i, e_j),\] and \[(g^{kl})_{1\le k, l\le 3}= g^{-1}.\] Moreover, denote \[g_{ij, k}= \partial_k g_{ij}.\] In local coordinates, Christoffel symbols are given by \[\Gamma_{ij}^k = \frac{1}{2} \sum_{m=1}^3 g^{km}(g_{mi,j} + g_{mj,i} - g_{ij,m}),\] and components of the Riemannian curvature tensor are given by \[{R_{lki}}^q = -\left(\partial_k \Gamma_{li}^q - \partial_l \Gamma_{ki}^q - \sum_{p=1}^3 \Gamma_{lp}^q \Gamma_{ki}^p + \sum_{p=1}^3 \Gamma_{kp}^q \Gamma_{li}^p\right).\] Denote \[\mathrm{Ric}_{ij}:= \mathrm{Ric}(e_i, e_j),\] which can be computed by \[\sum_{k=1}^3 g( \mathcal{R}(e_k, e_i)e_j, e_k )= \sum_{k=1}^3 \sum_{l} g(R_{kij}^{\;\;\;l} e_l, e_k)= \sum_k R_{kij}^{\;\;\;k}.\] Next, let us compute the first covariant derivative \[(\nabla_Z \mathrm{Ric})(X, Y),\] which, by definition, is equal to \[Z(\mathrm{Ric}(X, Y))- \mathrm{Ric}(\nabla_Z X, Y)- \mathrm{Ric}(X, \nabla_Z Y).\] Denote \[\mathrm{Ric}_{ij; k}:= (\nabla_{e_k} \mathrm{Ric})(e_i, e_j).\] In local coordinates, we can compute that \[\mathrm{Ric}_{ij; k}= \nabla_k \text{Ric}_{ij} = \partial_k \text{Ric}_{ij} - \sum_{m=1}^3 \Gamma_{ki}^m \text{Ric}_{mj} - \sum_{m=1}^3 \Gamma_{kj}^m \text{Ric}_{im}.\] In the end, we compute the second covariant derivative \[(\nabla_Z \nabla_Z \text{Ric})(X, Y).\] Denote \[\mathrm{Ric}_{ij; kl}:= (\nabla_{e_l}\nabla_{e_k} \mathrm{Ric})(e_i, e_j).\] We can compute that \[\begin{gather} \mathrm{Ric}_{ij; kl}=\partial_l \text{Ric}_{ij; k} \\ - \sum_{m=1}^3 \Gamma_{lk}^m \text{Ric}_{ij; m} - \sum_{m=1}^3 \Gamma_{li}^m \text{Ric}_{mj; k} - \sum_{m=1}^3 \Gamma_{lj}^m \text{Ric}_{im; k}. \end{gather}\] So far we have finished writing down the quantities we need to compute. Our goal is to find a metric \(g\) such that no matter which orthonormal basis \((X, Y, Z)\) at the origin we pick, the three quantities \[\mathrm{Ric}(X, Y), \;(\nabla_Z \mathrm{Ric})(X, Y), \;(\nabla_Z\nabla_Z \mathrm{Ric})(X, Y)\] never vanish at the same time.
We will work with the metric \[g=g_{11} d x^2+2 g_{12} d x d y+g_{22} d y^2+d z^2\] where \[\begin{gather} g_{11}=1-y^2+2 x y z-\frac{2}{3} x y^2 z-\frac{2}{9} x z^3+\frac{1}{3} y^4-\frac{3}{5} y^2 z^2-\frac{1}{3} z^4 \\ g_{12}=-\frac{1}{3} z^3-\frac{1}{3} x^2 y z+\frac{4}{5} x y z^2-\frac{1}{3} y^3 z-\frac{1}{9} y z^3 \\ g_{22}=1+z^2. \end{gather}\] To simplify our notation, let us denote \[A(X, Y)= \mathrm{Ric}(X, Y),\] \[B(Z, X, Y)= (\nabla_Z \mathrm{Ric})(X, Y),\] and \[C(W, Z, X, Y)= (\nabla_W \nabla_Z \mathrm{Ric})(X, Y)\] where everything is evaluated at the origin. Moreover, denote \[B_j(X, Y)= B(e_j, X, Y), \;j=1, 2, 3,\] and \[C_{jk}(X, Y)= C(e_j, e_k, X, Y), \;1\le j, k\le 3.\] By a direct calculation, we obtain \[A=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array}\right)\] \[B_1=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{array}\right), \quad B_2=0, \quad B_3=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right),\] and \[\begin{gather} C_{11}=\left(\begin{array}{ccc} 0 & 0 & -\frac{1}{3} \\ 0 & 0 & 0 \\ -\frac{1}{3} & 0 & 0 \end{array}\right), \quad C_{12}=\left(\begin{array}{ccc} 0 & -\frac{4}{5} & 0 \\ -\frac{4}{5} & 0 & \frac{1}{3} \\ 0 & \frac{1}{3} & 0 \end{array}\right), \\ C_{13}=\left(\begin{array}{ccc} \frac{2}{3} & 0 & \frac{4}{5} \\ 0 & 0 & 0 \\ \frac{4}{5} & 0 & \frac{2}{3} \end{array}\right), \\ C_{21}=\left(\begin{array}{ccc} 0 & \frac{1}{5} & 0 \\ \frac{1}{5} & 0 & \frac{1}{3} \\ 0 & \frac{1}{3} & 0 \end{array}\right), \quad C_{22}=\left(\begin{array}{ccc} \frac{6}{5} & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & \frac{6}{5} \end{array}\right), \\ C_{23}=\left(\begin{array}{ccc} 0 & \frac{1}{3} & 0 \\ \frac{1}{3} & 0 & 1 \\ 0 & 1 & 0 \end{array}\right), \\ C_{31}=\left(\begin{array}{ccc} \frac{2}{3} & 0 & \frac{4}{5} \\ 0 & 0 & 0 \\ \frac{4}{5} & 0 & \frac{2}{3} \end{array}\right), \quad C_{32}=\left(\begin{array}{ccc} 0 & \frac{1}{3} & 0 \\ \frac{1}{3} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right), \\ C_{33}=\left(\begin{array}{ccc} \frac{24}{5} & 0 & -\frac{1}{3} \\ 0 & \frac{24}{5} & 0 \\ -\frac{1}{3} & 0 & 8 \end{array}\right) . \end{gather}\] We will show that no matter which orthonormal basis \((X, Y, Z)\) we pick, the three quantities \[A(X, Y), \;B(Z, X, Y), \;C(Z, Z, X, Y)\] never vanish at the same time.
We introduce three auxiliary quantities \[\begin{gather} q_A=A(X, X)-A(Y, Y)+2 i A(X, Y), \\ q_B=B(Z, X, X)-B(Z, Y, Y)+2 i B(Z, X, Y), \\ q_C=C(Z, Z, X, X)-C(Z, Z, Y, Y)+2 i C(Z, Z, X, Y) . \end{gather}\] If it happens that \[A(X, Y)=B(Z, X, Y)=C(Z, Z, X, Y)=0,\] then the three quantities \(q_A, q_B, q_C\) are all real, and the three new quantities \[D_0=\operatorname{Im}\left(q_B \overline{q_C}\right), \quad D_1=-\operatorname{Im}\left(q_A \overline{q_C}\right), \quad D_2=\operatorname{Im}\left(q_A \overline{q_B}\right)\] must vanish at the same time.

Recall that our goal is to show that no matter which orthonormal basis \((X, Y, Z)\) we pick, the three quantities \[D_0, D_1, D_2\] cannot vanish at the same time. Write \[Z=(a, b, c), \;\;a^2+ b^2+ c^2=1.\] Let us first work with the case that \[r=\sqrt{a^2+ c^2}>0.\] By Claim 15, we can pick one fixed pair \((X, Y)\). Let us pick \[X=\left(\frac{c}{r}, 0,-\frac{a}{r}\right), \quad Y=\left(-\frac{a b}{r}, r,-\frac{b c}{r}\right) .\] Direct calculation shows that \[D_0= 4 a c(1+ b^2), \;D_1= 2b(1+ b^2), \;D_2= 2(1+ b^2)(a^2-c^2).\] It is elementary to see that these three quantities cannot vanish at the same time.
In the end, we still need to consider the case \(r=0\). Similar calculation shows that \[D_0=0,\; |D_1|=4, \;D_2=0.\] This finishes the proof of all the cases, and thus the proof that the metric we constructed satisfies the chaotic curvature condition of order \(\le 2\) at the origin along geodesics pointing in every direction. By continuity and compactness, we see that the chaotic curvature condition of order \(\le 2\) holds for neighboring points of the origin as well, and for small smooth perturbations of the metric as well.
Next, we will find a metric which fails the chaotic curvature condition of order \(\le 2\), and so does every small smooth perturbation of it. It is not difficult to imagine that there are plenty of these examples. More precisely, if an orthonormal frame \(X, Y, Z\) is a “transverse" solution to \[\mathrm{Ric}(X, Y)=(\nabla_Z \mathrm{Ric})(X, Y)=(\nabla_Z\nabla_Z \mathrm{Ric})(X, Y)=0\] at the origin, then the underlying metric works for our purpose. Here”transverse" means roughly that implicit function theorems can be applied to find solutions that are small perturbations of the frame \(X, Y, Z\). Let us carry out the details.
A very clean example is a conformal perturbation of the Euclidean metric. Denote \[u(x, y, z)=\frac{1}{2}\left(-x^2+z^2\right)-\frac{1}{2} y z^2+\frac{1}{6} x z^3,\] and consider the metric \[\label{260602e79587} g= e^{2 u(x, y, z)} (d x^2+ d y^2+ d z^2).\tag{75}\] First of all, one can repeat the calculation in the first half of this subsection, and show that if we take \[\label{260602e79588} (X, Y, Z)=(e_1, e_2, e_3),\tag{76}\] then \[\mathrm{Ric}(X, Y)=(\nabla_Z \mathrm{Ric})(X, Y)=(\nabla_Z\nabla_Z \mathrm{Ric})(X, Y)=0\] at the origin. It remains to show that this zero point is “transverse", whose meaning will become clear later. We perturb the zero point 76 , and consider a new orthogonal frame given by \[\label{260602e79590} \begin{align} & X= e_1+ \varepsilon_1 e_2+ \varepsilon_2 e_3+ O(\varepsilon^2), \\ & Y= e_2- \varepsilon_1 e_1+ \varepsilon_3 e_3+ O(\varepsilon^2)\\ & Z= e_3- \varepsilon_2 e_1- \varepsilon_3 e_2+ O(\varepsilon^2), \end{align}\tag{77}\] where \[\varepsilon^2:= \varepsilon_1^2+ \varepsilon_2^2+\varepsilon_3^2.\] Define the map \[\mathfrak{F}(\varepsilon_1, \varepsilon_2, \varepsilon_3)= \pnorm{ A(X, Y), B(Z, X, Y), C(Z, Z, X, Y) },\] where \(X, Y, Z\) are given by 77 . One can compute directly that \[\mathfrak{F}(\varepsilon_1, \varepsilon_2, \varepsilon_3)= \pnorm{ -\varepsilon_1+ O(\varepsilon^2), \varepsilon_2+ O(\varepsilon^2), 2\varepsilon_1- \varepsilon_3+ O(\varepsilon^2) }.\] Note that \[D \mathfrak{F}(0, 0, 0)= \begin{bmatrix} -1 & 0 & 0\\ 0 & 1 & 0\\ 2 & 0 & -1 \end{bmatrix}\] and the determinant of \(D\mathfrak{F}\) does not vanish. This is what we meant by that the zero point 76 is transverse. The rest of the proof is immediate: If we consider a small smooth perturbation of the metric 75 , then the new metric fails the chaotic curvature condition of order \(\le 2\) by the implicit function theorem. This finishes the proof of the second part of Item (2) in Theorem 4.

7.3 Proof of the third item↩︎

The proof is local, so we work in a neighborhood of the origin in \(\mathbb{R}^3\). The main point is that the failure of chaotic curvature of order \(\leq 3\) at a fixed point is detected by the \(5\)-jet of the metric at that point. Recall first why \(5\)-jets are enough. The Ricci tensor contains two derivatives of the metric. Therefore \(\nabla^m\operatorname{Ric}\) at a point is determined by derivatives of the metric up to order \(m+2\). In particular, \[\operatorname{Ric},\;\; \nabla\operatorname{Ric},\;\; \nabla^2\operatorname{Ric},\;\; \nabla^3\operatorname{Ric}\] at the origin are determined by the \(5\)-jet of the metric at the origin.
Let \(G^5_0\) denote the space of \(5\)-jets at the origin of Riemannian metrics on \(\mathbb{R}^3\). Thus \(G^5_0\) is an open subset of a finite-dimensional vector space. For \(\mathfrak g\in G^5_0\), and for a \(\mathfrak g(0)\)-orthonormal frame \((X,E,Z)\) of \(T_0\mathbb{R}^3\), define \[\mathcal{R}_m(\mathfrak g,X,E,Z) := (\nabla^m\operatorname{Ric})_{\mathfrak g,0} (\underbrace{Z,\ldots,Z}_{m},X,E), \qquad 0\leq m\leq 3.\] For \(m=0\), this means \[\mathcal{R}_0(\mathfrak g,X,E,Z) = \operatorname{Ric}_{\mathfrak g,0}(X,E).\] Define the bad set \(\Sigma_0\subset G^5_0\) by \[\begin{align} \Sigma_0 := \bigl\{ \mathfrak g\in G^5_0:\, &\text{there exists a \(\mathfrak g(0)\)-orthonormal frame } (X,E,Z) \\ &\text{such that } \mathcal{R}_m(\mathfrak g,X,E,Z)=0,\quad 0\leq m\leq 3 \bigr\}. \end{align}\] Thus \(\mathfrak g\notin\Sigma_0\) means that, for every orthonormal frame \((X,E,Z)\), at least one of \[\operatorname{Ric}(X,E),\;\; (\nabla_Z\operatorname{Ric})(X,E),\;\; (\nabla_Z^2\operatorname{Ric})(X,E),\;\; (\nabla_Z^3\operatorname{Ric})(X,E)\] is nonzero at the origin.

We claim that \(G^5_0\setminus\Sigma_0\) is open and dense in \(G^5_0\). The openness is straightforward by continuity and compactness. It remains to prove density. Consider the frame bundle \[\mathcal{B} := \bigl\{ (\mathfrak g,X,E,Z):\mathfrak g\in G^5_0,\, (X,E,Z)\text{ is \(\mathfrak g(0)\)-orthonormal} \bigr\}.\] If \(N=\dim G^5_0\), then \[\dim \mathcal{B}=N+3,\] because the space of orthonormal frames in dimension three has dimension \(3\). Define \[\mathcal{R}:\mathcal{B}\to\mathbb{R}^4, \qquad \mathcal{R}:=(\mathcal{R}_0,\mathcal{R}_1,\mathcal{R}_2,\mathcal{R}_3).\] It is not difficult to see that \(\mathcal{R}\) is a submersion, as the entries of \(\mathcal{R}\) involve derivatives of the metric of different degrees. So, by the submersion theorem, or equivalently the implicit function theorem, we obtain that \[\mathcal{Z}:=\mathcal{R}^{-1}(0)\] is a smooth submanifold of \(\mathcal{B}\) of codimension \(4\). In other words, \[\dim\mathcal{Z}=(N+3)-4=N-1.\] The bad set \(\Sigma_0\) is the projection of \(\mathcal{Z}\) to \(G^5_0\). Since all the defining relations are algebraic in the finite jet coefficients and in the frame variables, \(\Sigma_0\) is semialgebraic. Therefore \[\dim\Sigma_0\leq \dim\mathcal{Z}=N-1.\] In particular, \(\Sigma_0\) has empty interior in \(G^5_0\). This proves that \(G^5_0\setminus\Sigma_0\) is dense. Once this is proven, we can use continuity and compactness again to finish the proof of item (3).
S. G. is partly supported by the Nankai Zhide Foundation, NSFC Grant No. 12426204, and the New Cornerstone Science Foundation.
Part of this work was carried out during Gong’s visit to Nankai University; he acknowledges their hospitality with gratitude. S. Dai would like to thank Ruixiang Zhang for the discussion on the chaotic curvature condition.

Part of this work, especially Subsection 7.2, was developed with the assistance of large language models, including Doubao-Seed-2.0-pro, ChatGPT Pro 5.5, and Gemini Ultra 3.1-pro, at different stages of the project. In particular, Doubao-Seed suggested the first correct example for a model problem related to the second item of Theorem 2.7. Subsequently, ChatGPT was used to search for complete examples of the same type. All examples and arguments appearing in Subsection 7.2 were independently checked by the authors, who take full responsibility for their correctness.

S. Dai, Center for Applied Mathematics and KL-AAGDM, Tianjin University, Tianjin, 300072, China

E-mail address: song.dai@tju.edu.cn

L. Gong, Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China

E-mail address: lwgong@math.cuhk.edu.hk

S. Guo, Chern Institute of Mathematics, LPMC and New Cornerstone Science Laboratory, Nankai University, Tianjin, 300071, China

E-mail address: shaomingguo2018@gmail.com

References↩︎

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1. In particular, exactly the same numerology as that in the proof of the polynomial Wolff axiom in Subsection 4.3 in [2] works.↩︎

2. In comparison, manifolds of constant sectional curvature have “most possible" symmetries.↩︎

3. Note that when \(k=4\), item (3.3) is precisely Sogge’s chaotic curvature condition [1], which we call chaotic curvature condition of order \(\le 1\) in Definition 4. Moreover, in [2] it is proven that under Sogge’s chaotic curvature condition, the determinant of the leading \(4\times 4\) matrix of 4 does not vanish if \(\epsilon>0\) is chosen small enough, see Theorem 2.5 there.↩︎

4. In other words, what “generic" means in [16] is much more restrictive than in Theorem 4.↩︎

5. Here we add a bar on top of \(\nabla\) just to distinguish \(\bar{\nabla}\) from \(\nabla\): Under this notation, \(\bar{\nabla} f\) is a vector field, and \(\nabla f\) is a one-form, where \(f\) is a function.↩︎

6. Below the statement of Theorem 3 we already recalled some of these calculations.↩︎

7. Regarding the formula ?? , products of \(R^{(i)}\) start to appear when \(k=5\), and we do not see these product terms in Proposition 4.↩︎

8. Indeed 59 can be proven via applying Cayley-Hamilton theorem to ?? ; if we go back and check the proof of Proposition 11 again, we will see that this is precisely how we obtained 58 , and therefore 59 .↩︎