1 Introduction↩︎

One of the central themes in modern differential geometry concerns the relationship between the intrinsic and extrinsic geometry of submanifolds. A pivotal step in formalizing this viewpoint was taken by Nash [1], whose celebrated isometric embedding theorem established that every Riemannian manifold can be isometrically immersed into a Euclidean space of sufficiently high dimension. This result provides a firm theoretical foundation for studying abstract Riemannian manifolds as submanifolds of ambient spaces, and it naturally raises the question of how intrinsic invariants of a submanifold — such as its sectional curvature, Ricci curvature, and scalar curvature — are constrained by extrinsic quantities, most notably the mean curvature vector and the squared norm of the second fundamental form. Establishing sharp inequalities that relate these intrinsic and extrinsic invariants has become one of the most active and fruitful problems in submanifold theory.

A landmark contribution in this direction was made by Chen [2], who derived an optimal upper bound for the Ricci curvature of a submanifold of a real space form in terms of the squared mean curvature. This inequality, now widely referred to as the Chen-Ricci inequality, marked the beginning of a rich line of research. Building on Chen’s work, Hong and Tripathi [3] extended the Chen-Ricci inequality to submanifolds of an arbitrary Riemannian manifold and carefully analyzed the equality cases. Further extensions were subsequently obtained for submanifolds of contact metric manifolds by Tripathi [4], for Lagrangian submanifolds of complex space forms by Deng [5] and Oprea [6], [7], and for submanifolds of generalized Sasakian space forms by Hong and Tripathi [8]. More recently, the Chen-Ricci inequality and an improved version thereof were established for Kulkarni-Nomizu tensor fields satisfying an algebraic Gauss equation by Tripathi [9], and the equality cases were discussed in full generality. For a comprehensive survey of recent developments in Chen-Ricci inequalities and related results, we refer to [10].

Around the same period, Hineva independently investigated curvature bounds for submanifolds of Riemannian manifolds. In [11], she established upper and lower bounds for the sectional curvature in terms of the mean curvature vector and the squared norm of the second fundamental form. In [12], she further announced, without proof, both an upper and a lower bound for the Ricci curvature of a submanifold of a Riemannian manifold. The upper bound coincides in nature with the Chen-Ricci inequality, while the lower bound — which has come to be called the Hineva inequality [13] — reads as follows: if \((M, g)\) is an \(n\)-dimensional submanifold of a Riemannian manifold \((\widetilde{M}, \widetilde{g})\), then for any unit vector \(X \in T^{1}_{p}M\), \[\label{eq:Hineva95classical} \mathrm{Ric}(X) \;\geq\; \widetilde{\mathrm{Ric}}_{T_p M}(X) + \frac{n-1}{n}\!\left(2n\left\| H \right\|^2 - \left\| \sigma \right\|^2 - (n-2)\sqrt{\frac{n\left\| H \right\|^2\!\left(\left\| \sigma \right\|^2 - n\left\| H \right\|^2\right)}{n-1}}\right),\tag{1}\] where \(H\) denotes the mean curvature vector and \(\sigma\) denotes the second fundamental form of \(M\) in \(\widetilde{M}\). A complete proof of 1 and a full characterization of its equality case were provided by Hineva in [14], where it was shown that equality holds at a point \(p \in M\) precisely when \(p\) is a quasi-umbilical point of the submanifold. Subsequently, Verma, Mihai, Mihai, and Tripathi [13] established both the Chen-Ricci inequality and the Hineva inequality in the general framework of Kulkarni-Nomizu tensor fields on Riemannian manifolds satisfying an algebraic Gauss equation, and derived applications to various classes of submanifolds of generalized Sasakian space forms, including Sasakian [15], cosymplectic, and Kenmotsu space forms. In a companion paper [16], the same authors extended these results to submanifolds of product generalized Sasakian space forms.

While the foregoing developments have enriched the theory of submanifolds in various ambient spaces under the Levi-Civita connection, there has been a growing and well-motivated interest in studying submanifold geometry with respect to more general linear connections. Among these, the semi-symmetric non-metric connection, introduced by Agashe and Chafle [17] and further explored by several authors [18], [19], has emerged as a geometrically significant generalization. In particular, Özgür and Mihai [20] established Chen inequalities for submanifolds of real space forms endowed with this connection, providing the first systematic study of curvature inequalities in this non-metric setting. The precise definition, curvature properties, and the fundamental equations of submanifold theory with respect to this connection are recalled in detail in Section 2.

Real space forms, that is, Riemannian manifolds of constant sectional curvature \(c\), which include Euclidean space \(\mathbb{R}^m\), the sphere \(S^m(c)\), and the real hyperbolic space \(H^m(c)\) constitute the most classical ambient spaces in submanifold geometry. The study of curvature inequalities for their submanifolds under the Levi-Civita connection has yielded a wealth of results, among which the Chen-Ricci inequality [2] is most prominent. However, when the ambient real space form is endowed with a semi-symmetric non-metric connection, the corresponding theory of curvature inequalities, and in particular the Hineva inequality, remains largely unexplored.

Motivated by the above considerations, the purpose of the present paper is to establish the Hineva inequality for submanifolds of a real space form \(\widetilde{N}^{n+p}(c)\) endowed with a semi-symmetric non-metric connection. Using the Gauss equation adapted to this connection, we derive a sharp lower bound for the Ricci curvature of the submanifold in terms of the mean curvature vector and the squared norm of the second fundamental form, both computed with respect to the semi-symmetric non-metric connection. We also provide a complete characterization of the equality case, showing that equality holds precisely at quasi-umbilical points. As consequences, we derive the Hineva inequality for several geometrically significant classes of submanifolds, including invariant, anti-invariant, slant, semi-invariant, and semi-slant submanifolds of real space forms with semi-symmetric non-metric connection.

The paper is organized as follows. In Section 2, we recall the essential background on semi-symmetric non-metric connections, their curvature properties, and the fundamental equations of submanifold theory in this setting, together with the key algebraic lemma. Section 3 is devoted to the main result, where we state and prove the Hineva inequality and discuss the equality case in detail. In Section 4, we present the resulting corollaries for invariant, anti-invariant, slant, semi-invariant, and semi-slant submanifolds.

2 Preliminaries↩︎

In this section, we collect the essential background material that will be used throughout the paper. We recall the notion of a semi-symmetric non-metric connection on a Riemannian manifold, its curvature properties, the fundamental equations of submanifold theory adapted to this connection.

Let \((\widetilde{N}, \widetilde{g})\) be an \((n+p)\)-dimensional Riemannian manifold equipped with a linear connection \(\widetilde{\nabla}\). The torsion tensor \(\widetilde{T}\) of \(\widetilde{\nabla}\) is defined by \[\label{eq:torsion} \widetilde{T}(\widetilde{X}, \widetilde{Y}) = \widetilde{\nabla}_{\widetilde{X}} \widetilde{Y} - \widetilde{\nabla}_{\widetilde{Y}} \widetilde{X} - [\widetilde{X}, \widetilde{Y}],\tag{2}\] for all vector fields \(\widetilde{X}\) and \(\widetilde{Y}\) on \(\widetilde{N}\). The connection \(\widetilde{\nabla}\) is called semi-symmetric if its torsion tensor satisfies \[\label{eq:semisymmetric} \widetilde{T}(\widetilde{X}, \widetilde{Y}) = \phi(\widetilde{Y})\widetilde{X} - \phi(\widetilde{X})\widetilde{Y},\tag{3}\] for some \(1\)-form \(\phi\) on \(\widetilde{N}\). Moreover, \(\widetilde{\nabla}\) is said to be metric if \(\widetilde{\nabla}\widetilde{g} = 0\), and non-metric if \(\widetilde{\nabla}\widetilde{g} \neq 0\). A connection that is simultaneously semi-symmetric and non-metric is called a semi-symmetric non-metric connection.

Following Agashe and Chafle [17], a semi-symmetric non-metric connection \(\widetilde{\nabla}\) on \(\widetilde{N}\) is expressed in terms of the Levi-Civita connection \(\mathring{\widetilde{\nabla}}\) of \(\widetilde{g}\) by \[\label{eq:connection95formula} \widetilde{\nabla}_{\widetilde{X}} \widetilde{Y} = \mathring{\widetilde{\nabla}}_{\widetilde{X}} \widetilde{Y} + \phi(\widetilde{Y})\widetilde{X},\tag{4}\] for all vector fields \(\widetilde{X}\) and \(\widetilde{Y}\) on \(\widetilde{N}\), where \(\phi\) is a \(1\)-form on \(\widetilde{N}\). The vector field \(P\) on \(\widetilde{N}\) associated with \(\phi\) is defined by \(\widetilde{g}(P, \widetilde{X}) = \phi(\widetilde{X})\) for all \(\widetilde{X} \in \chi(\widetilde{N})\).

Let \(\widetilde{R}\) and \(\mathring{\widetilde{R}}\) denote the curvature tensors of \(\widetilde{N}\) with respect to \(\widetilde{\nabla}\) and \(\mathring{\widetilde{\nabla}}\), respectively. According to [17], the two curvature tensors are related by \[\label{eq:curvature95relation} \widetilde{R}(X, Y, Z, W) = \mathring{\widetilde{R}}(X, Y, Z, W) + s(X, Z)\,\widetilde{g}(Y, W) - s(Y, Z)\,\widetilde{g}(X, W),\tag{5}\] for all vector fields \(X, Y, Z, W\) on \(\widetilde{N}\), where \(s\) is the \((0,2)\)-tensor field on \(\widetilde{N}\) defined by \[\label{eq:s95tensor} s(X, Y) = \left(\mathring{\widetilde{\nabla}}_{X}\,\phi\right)Y - \phi(X)\phi(Y), \qquad X, Y \in \chi(\widetilde{N}).\tag{6}\] We denote by \(\lambda\) the trace of \(s\), that is, \(\lambda = \sum_{i=1}^{n} s(e_i, e_i)\) for any local orthonormal frame \(\{e_1, \ldots, e_n\}\).

Let \(\widetilde{N}^{n+p}(c)\) denote a real space form of constant sectional curvature \(c\), that is, a Riemannian manifold whose Levi-Civita curvature tensor satisfies \[\label{eq:RSF95Levi} \mathring{\widetilde{R}}(X, Y, Z, W) = c\left\{\widetilde{g}(X, W)\widetilde{g}(Y, Z) - \widetilde{g}(X, Z)\widetilde{g}(Y, W)\right\}.\tag{7}\] Substituting 7 into 5 , the curvature tensor of \(\widetilde{N}^{n+p}(c)\) with respect to the semi-symmetric non-metric connection \(\widetilde{\nabla}\) takes the form \[\label{eq:RSF95curvature} \widetilde{R}(X, Y, Z, W) = c\left\{\widetilde{g}(X, W)\widetilde{g}(Y, Z) - \widetilde{g}(X, Z)\widetilde{g}(Y, W)\right\} + s(X, Z)\,\widetilde{g}(Y, W) - s(Y, Z)\,\widetilde{g}(X, W).\tag{8}\]

Let \((M^n, g)\) be an \(n\)-dimensional Riemannian submanifold of the real space form \(\widetilde{N}^{n+p}(c)\) endowed with the semi-symmetric non-metric connection \(\widetilde{\nabla}\). We denote by \(\nabla\) the connection on \(M^n\) induced from \(\widetilde{\nabla}\), and by \(\mathring{\nabla}\) the Levi-Civita connection of \((M^n, g)\).

The Gauss formulas with respect to \(\widetilde{\nabla}\) and \(\mathring{\widetilde{\nabla}}\) are given respectively by \[\begin{align} \widetilde{\nabla}_{X} Y &= \nabla_{X} Y + h(X, Y), \qquad X, Y \in \chi(M^n), \tag{9}\\ \mathring{\widetilde{\nabla}}_{X} Y &= \mathring{\nabla}_{X} Y + \mathring{h}(X, Y), \qquad X, Y \in \chi(M^n), \tag{10} \end{align}\] where \(\mathring{h}\) is the second fundamental form of \(M^n\) in \(\widetilde{N}^{n+p}(c)\) with respect to the Levi-Civita connection, and \(h\) is the corresponding \((0,2)\)-tensor with respect to the semi-symmetric non-metric connection. According to [21], these two tensors coincide: \[\label{eq:h95equal} h = \mathring{h}.\tag{11}\] In view of 11 , we denote the common second fundamental form simply by \(\sigma\), and write the unified Gauss formula as \[\label{eq:Gauss95unified} \widetilde{\nabla}_{X} Y = \nabla_{X} Y + \sigma(X, Y), \qquad X, Y \in \chi(M^n).\tag{12}\]

The mean curvature vector \(H\) of \(M^n\) in \(\widetilde{N}^{n+p}(c)\) is defined by \[\label{eq:mean95curvature} H = \frac{1}{n}\,\mathrm{trace}\,\sigma = \frac{1}{n}\sum_{i=1}^{n} \sigma(e_i, e_i),\tag{13}\] and the squared norm of the second fundamental form is \[\label{eq:sigma95norm} \|\sigma\|^2 = \sum_{\alpha=n+1}^{n+p}\sum_{i,j=1}^{n} \left(\sigma_{ij}^{\alpha}\right)^2,\tag{14}\] where \(\sigma_{ij}^{\alpha} = \widetilde{g}(\sigma(e_i, e_j), e_\alpha)\) for an orthonormal frame \(\{e_1, \ldots, e_n\}\) of \(TM^n\) and an orthonormal frame \(\{e_{n+1}, \ldots, e_{n+p}\}\) of the normal bundle \(T^{\perp}M^n\). We also write \(H^{\alpha} = \frac{1}{n}\sum_{i=1}^{n}\sigma_{ii}^{\alpha}\), so that \(\|H\|^2 = \sum_{\alpha}(H^{\alpha})^2\).

The vector field \(P\) on \(\widetilde{N}^{n+p}(c)\) decomposes uniquely into its tangential and normal components along \(M^n\) as \[\label{eq:P95decomp} P = P^{\top} + P^{\perp},\tag{15}\] where \(P^{\top} \in TM^n\) and \(P^{\perp} \in T^{\perp}M^n\). We adopt the notation \(\phi(H) = \widetilde{g}(P^{\perp}, H)\).

The Gauss equation for \(M^n\) in \(\widetilde{N}^{n+p}(c)\) with respect to the semi-symmetric non-metric connection \(\widetilde{\nabla}\) is given by [21]: \[\begin{align} \label{eq:Gauss95equation} \widetilde{R}(X, Y, Z, W) &= R(X, Y, Z, W) + \widetilde{g}(\sigma(X,Z), \sigma(Y,W)) - \widetilde{g}(\sigma(Y,Z), \sigma(X,W)) \notag \\ &\quad + \widetilde{g}(P^{\perp}, \sigma(Y,Z))\,\widetilde{g}(X,W) - \widetilde{g}(P^{\perp}, \sigma(X,Z))\,\widetilde{g}(Y,W), \end{align}\tag{16}\] where \(R\) is the curvature tensor of \(M^n\) with respect to the induced connection \(\nabla\).

For any point \(x \in M^n\) and any orthonormal basis \(\{e_1, \ldots, e_n\}\) of \(T_x M^n\), the scalar curvature \(\tau\) at \(x\) is defined by \[\label{eq:scalar95curvature} \tau(x) = \sum_{1 \leq i < j \leq n} K(e_i \wedge e_j),\tag{17}\] where \(K(e_i \wedge e_j)\) denotes the sectional curvature of \(M^n\) with respect to \(\nabla\) in the direction of the \(2\)-plane spanned by \(e_i\) and \(e_j\).

A submanifold \(M^n\) is said to be totally geodesic if \(\sigma \equiv 0\), and minimal if \(H = 0\). A point \(x \in M^n\) is called an umbilical point if the shape operator satisfies \(A_{N} = \rho(N)\,\mathrm{Id}\) for all normal vectors \(N \in T^{\perp}_x M^n\) and some scalar \(\rho(N)\). The submanifold \(M^n\) is totally umbilical if every point is umbilical. Furthermore, a point \(x \in M^n\) is called quasi-umbilical with respect to a normal direction \(e_\alpha\) if the matrix \((\sigma_{ij}^\alpha)\) has exactly two distinct eigenvalues \(\lambda_\alpha\) and \(\mu_\alpha\), with multiplicities \(1\) and \(n-1\), respectively [14].

The following algebraic lemma, due to Hineva [14], is the principal tool in establishing the lower bound for the Ricci curvature.

3 Main Result↩︎

4 Applications to Special Submanifolds↩︎

In this section, we apply Theorem 1 to several classes of submanifolds. Throughout, \(M^n\) denotes an \(n\)-dimensional submanifold of the real space form \(\widetilde{N}^{n+p}(c)\) endowed with the semi-symmetric non-metric connection \(\widetilde{\nabla}\), and \(X \in T^1_x M^n\) is an arbitrary unit tangent vector at a point \(x \in M^n\). For convenience, we denote \[\label{eq:Hcal} \mathcal{H}(H,\sigma) = \frac{n-1}{n} \left( 2n\|H\|^2 - \|\sigma\|^2 - (n-2)\sqrt{\frac{n\|H\|^2 \left(\|\sigma\|^2 - n\|H\|^2\right)}{n-1}} \right).\tag{23}\]

References↩︎