Harmonic and Killing Vector Fields in Compact Orientable Riemannian Spaces with Boundary

Abstract

Publisher Summary This chapter presents the results on harmonic and killing vector fields to the case of Riemannian spaces with boundary. The harmonic tensors on such Riemannian spaces are studied, and the relations between curvature and relative Betti numbers in Riemannian spaces with boundary are discussed. If, in an M with boundary B, the Ricci curvature is positive (negative) definite and the second fundamental form is negative (positive) semi-definite, then there does not exist a harmonic (Killing, conformal Eilling) vectoy field tangential to B other than the zero vector field. If, in an M with boundary B, the Ricci curvature is positive (negative) definite and the mean curvature of B is negative (positive) or zero, then there does not exist a harmonic (Killing, conformal Killing) vector field normal to B other than the zero vector field. If an M admits a Killing vector field normal to the boundary B and vanishing only on a nowhere dense set of points on B, then the boundary is totally geodesic.