Hypersurfaces of a Sasakian Manifold

Abstract

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound.

Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

“…Some inequalities for the length of the second fundamental form are obtained, including the length of warping functions and slant immersions. Various inequalities, which have been derived in [13,14,[18][19][20][21][22][23][24][25][26][27], can be recovered from our inequalities under some conditions. Therefore, our results may find applications in mathematical physics.…”

On Killing Vector Fields on Riemannian Manifolds

“…Killing vector fields are known to play vital role in influencing the geometry as well as topology of Riemannian manifolds (see [1][2][3][4][5][6][7][8][9][10]) and being incompressible fields play important role in physics (see [11]). If we restrict the length of a Killing vector fields such as constant length, then it severely restricts the geometry of Riemannian manifolds on which they are set.…”

Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

“…erefore, the characterization of these spaces the Euclidean space R n , the Euclidean sphere S n , and the complex projective space CP n are recognized fields in the study of differential geometry and are studied in research works such as [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In particular, the Euclidean space R n is designated through the differential equation ∇ 2 φ � cg, where c is a positive constant, which is proven by Tashiro [19].…”