Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections
© Lee et al.; licensee Springer. 2014
Received: 17 May 2014
Accepted: 30 July 2014
Published: 1 September 2014
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 semi-symmetric metric connection. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds.
The idea of a semi-symmetric linear connection on a differentiable manifold was introduced by Friedmann and Schouten in . The notion of a semi-symmetric metric connection on a Riemannian manifold was introduced by Hayden in . Later, Yano in  studied some properties of a Riemannian manifold endowed with a semi-symmetric metric connection. In [4, 5], Imai found some properties of a Riemannian manifold and a hypersurface of a Riemannian manifold with a semi-symmetric metric connection. Nakao in  studied submanifolds of a Riemannian manifold with semi-symmetric metric connections.
On the other hand, the theory of Chen invariants, initiated by Chen  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 . 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 semi-symmetric 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 semi-symmetric metric connections. Moreover, Zhang et al.  obtained Chen-like inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature endowed with a semi-symmetric 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  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 . 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 semi-symmetric metric connections.
- (i)The normalized δ-Casorati curvature satisfiesMoreover, the equality sign holds if and only if is an invariantly quasi-umbilical submanifold with trivial normal connection in , such that with respect to suitable orthonormal tangent frame and normal orthonormal frame , the shape operators , , take the following forms:
- (ii)The normalized δ-Casorati curvature satisfiesMoreover, the equality sign holds if and only if is an invariantly quasi-umbilical submanifold with trivial normal connection in , such that with respect to suitable orthonormal tangent frame and normal orthonormal frame , the shape operators , , take the following forms:
for a 1-form ϕ, then the connection is called a semi-symmetric connection.
Let g be a Riemannian metric on . If , then is called a semi-symmetric metric connection on .
for any vector fields and on , where denotes the Levi-Civita connection with respect to the Riemannian metric g and P is a vector field defined by , for any vector field .
We will consider a Riemannian manifold endowed with a semi-symmetric metric connection and the Levi-Civita connection denoted by .
Let be an n-dimensional submanifold of an m-dimensional Riemannian manifold . On the submanifold , we consider the induced semi-symmetric metric connection, denoted by ∇ and the induced Levi-Civita connection, denoted by .
Let be the curvature tensor of with respect to and the curvature tensor of with respect to . We also denote by R and the curvature tensors of ∇ and , respectively, on .
where is the second fundamental form of in and h is a -tensor on . According to the formula (7) from , h is also symmetric. One denotes by the mean curvature vector of and . Let be a real space form of constant sectional curvature c endowed with a semi-symmetric metric connection .
Denote by λ the trace of α.
The submanifold M is called invariantly quasi-umbilical if there exist mutually orthogonal unit normal vectors such that the shape operators with respect to all directions have an eigenvalue of multiplicity and that for each the distinguished eigendirection is the same .
3 Proof of the theorem
for every tangent hyperplane L of M. Taking the infimum over all tangent hyperplane L, the theorem trivially follows.
Remark We have a slightly modified coefficient in the definition of ; in fact, it was used the coefficient , as in [22, 23, 25], instead of , like in the present paper because we are working on the generalized normalized δ-Casorati curvature for a positive real number , as in .
The authors would like to thank the referee for his valuable comments and suggestions which helped to improve the paper.
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