Characterizing small spheres in a unit sphere by Fischer–Marsden equation
Characterizing small spheres in a unit sphere by Fischer–Marsden equation
We use a nontrivial concircular vector field u on the unit sphere Sn+1 in studying
geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere Sn+1
naturally inherits a vector field v and a smooth function ρ. We use the condition that
the vector field v is an eigenvector of the de-Rham Laplace operator together with an
inequality satisfied by the integral of the Ricci curvature in the direction of the vector
field v to find a characterization of small spheres in the unit sphere Sn+1. We also use
the condition that the function ρ is a nontrivial solution of the Fischer–Marsden
equation together with an inequality satisfied by the integral of the Ricci curvature in
the direction of the vector field v to find another characterization of small spheres in
the unit sphere Sn+1.
We aim to examine the influence of the existence of a nonzero eigenvector z of the de-Rham
operator G on a k-dimensional Riemannian manifold (Nk, g). If the vector z annihilates the de-Rham…
The Fischer-Marsden conjecture asserts that an n-dimensional compact manifold admitting a nontrivial solution of the so-called Fischer-Marsden differential equation is necessarily an Einstein…
We use a nontrivial concircular vector field u on the unit sphere Sn+1 in studying
geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere Sn+1
naturally…