Abstract
Abstract
In the present paper, we investigate the geometry and topology of warped product submanifolds in Riemannian warped product
I
×
f
S
m
(
c
)
and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Our main object is to apply geometrically to number structures and obtained applications of Dirichlet eigenvalues problems. Some new results on mean curvature vanishing are presented, and a partial solution can be found to the well-known problem given by S S Chern. The family of Riemannian warped products generalized to Robertson-Walker space-times
Funder
Deputyship of Scientific Research, Princess Nourah Bint Abdulrahman University
Reference34 articles.
1. The embedding problem for Riemannian manifolds;Nash;Ann. Math.,1956
2. Mean curvature and shape operator of isometric immersions in real space forms;Chen;Glasgow Math. J.,1996
3. Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension;Chen;Glasgow Math. J.,1999
4. Characterization of Riemannian space forms, Einstein spaces, and conformally flat spaces;Chen;Proc. Amer. Math. Soc.,1999
5. Isometric immersions of Riemannian manifolds;Gromov;The Mathematical Heritage of Élie Cartan, Lyon, 1984, Astérisque,1985