Vector fields and infinitesimal transformations on riemannian manifolds with boundary

TL;DR: In this article, the authors extended the results of studies made of vector fields or infinitesimal transformations on compact Riemannian manifolds without boundary, and they extended these results to Riemmannian manifold with boundary and gave necessary and sufficient conditions for a vector field on a manifold with zero tangential or normal component on the boundary to be a killing vector field.

Abstract: : Results of studies made of vector fields or infinitesimal transformations on compact Riemannian manifolds without boundary are extended to Riemannian manifolds with boundary. Fundamental formulas for Lie derivatives are given and the infinitesimal transformations and their generating vector fields are defined in terms of Lie derivatives. Necessary and sufficient conditions for a vector field on a manifold with zero tangential or normal component on a boundary to be a killing vector field are given. Conditions are obtained for the nonexistence of a nonzero conformal killing vector field on a manifold with zero tangential or normal component on the boundary, and necessary and sufficient conditions for a vector field on a manifold with zero tangential or normal component on the boundary to be a conformal killing vector field are obtained. It is shown that if the manifold has constant scalar curvature and admits a certain special infinitesimal nonhomothetic conformal motion leaving the boundary invariant, then the curvature is greater than zero. On a compact orientible Einstein manifold with the same boundary and curvature greater than zero, those special infinitesimal nonhomothetic conformal motions leaving the boundary invariant form a Lie algebra; a decomposition of this algebra with interrelations between its subalgebras is also obtained.