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Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semisymmetric metric connections
Journal of Inequalities and Applications volume 2014, Article number: 327 (2014)
Abstract
In this paper, we prove two optimal inequalities involving the intrinsic scalar curvature and extrinsic Casorati curvature of submanifolds of real space forms endowed with a semisymmetric metric connection. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasiumbilical submanifolds.
MSC:53C40, 53B05.
1 Introduction
The idea of a semisymmetric linear connection on a differentiable manifold was introduced by Friedmann and Schouten in [1]. The notion of a semisymmetric metric connection on a Riemannian manifold was introduced by Hayden in [2]. Later, Yano in [3] studied some properties of a Riemannian manifold endowed with a semisymmetric metric connection. In [4, 5], Imai found some properties of a Riemannian manifold and a hypersurface of a Riemannian manifold with a semisymmetric metric connection. Nakao in [6] studied submanifolds of a Riemannian manifold with semisymmetric metric connections.
On the other hand, the theory of Chen invariants, initiated by Chen [7] in a seminal paper published in 1993, is presently one of the most interesting research topic in differential geometry of submanifolds. Chen established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature, and the squared mean curvature. That is, he established simple relationships between the main intrinsic invariants and the main extrinsic invariants of a submanifold in real space forms with any codimensions in [8]. Many famous results concerned Chen invariants and inequalities for the different classes of submanifolds in various ambient spaces, like complex space forms [9–11]. Recently, in [12, 13], Mihai and Özgür proved Chen inequalities for submanifolds of real, complex, and Sasakian space forms endowed with semisymmetric metric connections and in [14, 15], Özgür and Murathan gave Chen inequalities for submanifolds of a locally conformal almost cosymplectic manifold and a cosymplectic space form endowed with semisymmetric metric connections. Moreover, Zhang et al. [16] obtained Chenlike inequalities for submanifolds of a Riemannian manifold of quasiconstant curvature endowed with a semisymmetric metric connection by using an algebraic approach.
Instead of concentrating on the sectional curvature with the extrinsic squared mean curvature, the Casorati curvature of a submanifold in a Riemannian manifold was considered as an extrinsic invariant defined as the normalized square of the length of the second fundamental form. The notion of Casorati curvature extends the concept of the principal direction of a hypersurface of a Riemannian manifold. Several geometers in [17–21] found geometrical meaning and the importance of the Casorati curvature. Therefore, it is of great interest to obtain optimal inequalities for the Casorati curvatures of submanifolds in different ambient spaces. Decu et al. in [22] obtained some optimal inequalities involving the scalar curvature and the Casorati curvature of a Riemannian submanifold in a real space form and the holomorphic sectional curvature and the Casorati curvature of a Kähler hypersurface in a complex space form. They also proved an inequality in which the scalar curvature is estimated from above by the normalized Casorati curvatures in [23]. Recently, some optimal inequalities involving Casorati curvatures were proved in [24, 25] for slant submanifolds in quaternionic space forms.
As a natural prolongation of our research, in this paper we will study these inequalities for submanifolds in real space forms, endowed with semisymmetric metric connections.
Theorem 1.1 Let {M}^{n} be a submanifold of a real space form {N}^{m}(c) with a semisymmetric metric connection. Then:

(i)
The normalized δCasorati curvature {\delta}_{C}(n1) satisfies
\rho \le {\delta}_{C}(n1)+c\frac{2}{n}trace(\alpha ).Moreover, the equality sign holds if and only if {M}^{n} is an invariantly quasiumbilical submanifold with trivial normal connection in {N}^{m}(c), such that with respect to suitable orthonormal tangent frame \{{\xi}_{1},\dots ,{\xi}_{n}\} and normal orthonormal frame \{{\xi}_{n+1},\dots ,{\xi}_{m}\}, the shape operators {A}_{r}\equiv {A}_{{\xi}_{r}}, r\in \{n+1,\dots ,m\}, take the following forms:
{A}_{n+1}=\left(\begin{array}{cccccc}a& 0& 0& \cdots & 0& 0\\ 0& a& 0& \cdots & 0& 0\\ 0& 0& a& \cdots & 0& 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& 0& \cdots & a& 0\\ 0& 0& 0& \cdots & 0& 2a\end{array}\right),\phantom{\rule{2em}{0ex}}{A}_{n+2}=\cdots ={A}_{m}=0. 
(ii)
The normalized δCasorati curvature {\stackrel{\u02c6}{\delta}}_{C}(n1) satisfies
\rho \le {\stackrel{\u02c6}{\delta}}_{C}(n1)+c\frac{2}{n}trace(\alpha ).Moreover, the equality sign holds if and only if {M}^{n} is an invariantly quasiumbilical submanifold with trivial normal connection in {N}^{m}(c), such that with respect to suitable orthonormal tangent frame \{{\xi}_{1},\dots ,{\xi}_{n}\} and normal orthonormal frame \{{\xi}_{n+1},\dots ,{\xi}_{m}\}, the shape operators {A}_{r}\equiv {A}_{{\xi}_{r}}, r\in \{n+1,\dots ,m\}, take the following forms:
{A}_{n+1}=\left(\begin{array}{cccccc}2a& 0& 0& \cdots & 0& 0\\ 0& 2a& 0& \cdots & 0& 0\\ 0& 0& 2a& \cdots & 0& 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& 0& \cdots & 2a& 0\\ 0& 0& 0& \cdots & 0& a\end{array}\right),\phantom{\rule{2em}{0ex}}{A}_{n+2}=\cdots ={A}_{m}=0.
2 Preliminaries
Let {N}^{m} be an mdimensional Riemannian manifold and \tilde{\mathrm{\nabla}} a linear connection on {N}^{m}. If the torsion tensor \tilde{T} of \tilde{\mathrm{\nabla}}, defined by
for any vector field \tilde{X} and \tilde{Y} on {N}^{m}, satisfies
for a 1form ϕ, then the connection \tilde{\mathrm{\nabla}} is called a semisymmetric connection.
Let g be a Riemannian metric on {N}^{m}. If \tilde{\mathrm{\nabla}}g=0, then \tilde{\mathrm{\nabla}} is called a semisymmetric metric connection on {N}^{m}.
Following [3], a semisymmetric metric connection \tilde{\mathrm{\nabla}} on {N}^{m} is given by
for any vector fields \tilde{X} and \tilde{Y} on {N}^{m}, where \stackrel{\u02da}{\tilde{\mathrm{\nabla}}} denotes the LeviCivita connection with respect to the Riemannian metric g and P is a vector field defined by g(P,\tilde{X})=\varphi (\tilde{X}), for any vector field \tilde{X}.
We will consider a Riemannian manifold {N}^{m} endowed with a semisymmetric metric connection \tilde{\mathrm{\nabla}} and the LeviCivita connection denoted by \stackrel{\u02da}{\tilde{\mathrm{\nabla}}}.
Let {M}^{n} be an ndimensional submanifold of an mdimensional Riemannian manifold {N}^{m}. On the submanifold {M}^{n}, we consider the induced semisymmetric metric connection, denoted by ∇ and the induced LeviCivita connection, denoted by \stackrel{\u02da}{\mathrm{\nabla}}.
Let \tilde{R} be the curvature tensor of {N}^{m} with respect to \tilde{\mathrm{\nabla}} and \stackrel{\u02da}{\tilde{R}} the curvature tensor of {N}^{m} with respect to \stackrel{\u02da}{\tilde{\mathrm{\nabla}}}. We also denote by R and \stackrel{\u02da}{R} the curvature tensors of ∇ and \stackrel{\u02da}{\mathrm{\nabla}}, respectively, on {M}^{n}.
The Gauss formulas with respect to ∇ and \stackrel{\u02da}{\mathrm{\nabla}}, respectively, can be written as
where \stackrel{\u02da}{h} is the second fundamental form of {M}^{n} in {N}^{m} and h is a (0,2)tensor on {M}^{n}. According to the formula (7) from [6], h is also symmetric. One denotes by \stackrel{\u02da}{H} the mean curvature vector of {M}^{n} and {N}^{m}. Let {N}^{m}(c) be a real space form of constant sectional curvature c endowed with a semisymmetric metric connection \tilde{\mathrm{\nabla}}.
The curvature tensor \stackrel{\u02da}{\tilde{R}} with respect to the LeviCivita connection \stackrel{\u02da}{\tilde{\mathrm{\nabla}}} on {N}^{m}(c) is expressed by
Then the curvature tensor \tilde{R} with respect to the semisymmetric metric connection \tilde{\mathrm{\nabla}} on {N}^{m}(c) can be written as [5]
for any vector fields X,Y,Z,W\in \chi ({M}^{n}), where α is a (0,2)tensor field defined by
From (2.1) and (2.2), it follows that the curvature tensor \tilde{R} can be expressed as
Denote by λ the trace of α.
The Gauss equation for the submanifold {M}^{n} in the real space form {N}^{m}(c) is
Let \pi \subset {T}_{x}{M}^{n}, x\in {M}^{n}, be a 2plane section. Denote by K(\pi ) the sectional curvature of {M}^{n} with respect to the induced semisymmetric metric connection ∇. For any orthonormal basis \{{e}_{1},\dots ,{e}_{n}\} of the tangent space {T}_{x}{M}^{n} and \{{e}_{n+1},\dots ,{e}_{m}\} is an orthonormal basis of the normal space {T}_{x}^{\mathrm{\perp}}M, then the scalar curvature τ at x is defined by
and the normalized scalar curvature ρ of M is defined by
We denote by H the mean curvature vector, that is,
and we also set
Then the squared mean curvature of the submanifold M in N is defined by
and the squared norm of h over dimension n is denoted by \mathcal{C} and is called the Casorati curvature of the submanifold M. Therefore, we have
The submanifold M is called invariantly quasiumbilical if there exist mn mutually orthogonal unit normal vectors {\xi}_{n+1},\dots ,{\xi}_{m} such that the shape operators with respect to all directions {\xi}_{\alpha} have an eigenvalue of multiplicity n1 and that for each {\xi}_{\alpha} the distinguished eigendirection is the same [26].
Suppose now that L is an rdimensional subspace of {T}_{x}M, r\ge 2, and \{{e}_{1},\dots ,{e}_{r}\} be an orthonormal basis of L. Then the scalar curvature \tau (L) of the rplane section L is given by
and the Casorati curvature \mathcal{C}(L) of the subspace L is defined as
The normalized δCasorati curvature {\delta}_{c}(n1) and {\stackrel{\u02c6}{\delta}}_{c}(n1) are given by
and
3 Proof of the theorem
From [6], the Gauss equation with respect to the semisymmetric metric connection is
Let x\in {M}^{n} and \{{e}_{1},{e}_{2},\dots ,{e}_{n}\} and \{{e}_{n+1},\dots ,{e}_{m}\} be orthonormal basis of {T}_{x}{M}^{n} and {T}_{x}^{\mathrm{\perp}}{M}^{n}, respectively. For X=W={e}_{i}, Y=Z={e}_{j}, i\ne j, from (2.3), it follows that
From (3.1) and (3.2), we get
By summation over 1\le i,j\le n, it follows from the previous relation that
We define now the following function, denoted by \mathcal{P}, which is a quadratic polynomial in the components of the second fundamental form:
Without loss of generality, by assuming that L is spanned by {e}_{1},\dots ,{e}_{n1}, one derives that
and now we easily obtain
From (3.4), it follows that the critical points
of \mathcal{P} are the solutions of the following system of linear homogeneous equations:
with i,j\in \{1,\dots ,n1\}, i\ne j and \alpha \in \{n+1,\dots ,m\}. Thus, every solution {h}^{c} has {h}_{ij}^{\alpha}=0 for i\ne j, and the determinant which corresponds to the first two sets of equations of the above system is zero (there exist solutions for nontotally geodesic submanifolds). Moreover, it is easy to see that the Hessian matrix of \mathcal{P} has the form
where
0 denotes the null matrix of corresponding dimensions and {H}_{2}, {H}_{3} are the next diagonal matrices
Therefore, we find that \mathcal{H}(\mathcal{P}) has the following eigenvalues:
Therefore, \mathcal{P} is parabolic and reaches a minimum \mathcal{P}({h}^{c})=0 for the solution {h}^{c} of the system (3.5). It follows that \mathcal{P}\ge 0, and, hence,
Hence, we deduce that
for every tangent hyperplane L of M. Taking the infimum over all tangent hyperplane L, the theorem trivially follows.
Moreover, we can easily check that the equality sign holds in the theorem if and only if
and
From (3.6) and (3.7), we conclude that the equality holds if and only if the submanifold M is invariantly quasiumbilical with trivial normal connection in N, such that with respect to suitable orthonormal tangent and normal orthonormal frames, the shape operators take the forms below:
Remark We have a slightly modified coefficient in the definition of {\delta}_{C}(n1); in fact, it was used the coefficient \frac{n+1}{2n(n1)}, as in [22, 23, 25], instead of \frac{n+1}{2n}, like in the present paper because we are working on the generalized normalized δCasorati curvature {\delta}_{C}(r;n1) for a positive real number r\ne n(n1), as in [24].
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Lee, C.W., Yoon, D.W. & Lee, J.W. Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semisymmetric metric connections. J Inequal Appl 2014, 327 (2014). https://doi.org/10.1186/1029242X2014327
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DOI: https://doi.org/10.1186/1029242X2014327
Keywords
 Casorati curvature
 real space form
 semisymmetric metric connection