Geometry of infinitesimal harmonic transformations (Q1403401)

The vector field \(\xi\) is an infinitesimal harmonic transformation in a Riemannian manifold \((M, g)\) if the local one-parameter group of infinitesimal point transformations generated by the vector field \(\xi\) is a group of harmonic transformations. The authors study the local and global geometry for infinitesimal harmonic transformations in a Riemannian manifold. A remarkable theorem proved by Lichnerowicz is: A holomorphic vector field \(\xi\) in a compact Kählerian manifold \((M, g, J)\) with constant scalar curvature is decomposed in the form \(\xi= \xi'+ J\xi''\) where \(\xi'\) and \(\xi''\) are both infinitesimal isometric transformations. The authors combine the results of Yano to prove the following theorem: A compact and orientable Riemannian manifold \((M, g)\) with negative definite Ricci curvature does not admit infinitesimal harmonic transformations. Moreover, they prove an interesting theorem as follows: Let \((M, g)\) be a compact Riemannian manifold. Then the vector space \(H(M, R)\) of infinitesimal harmonic transformations in \((M, g)\) is finite-dimensional, and \(\dim H(M, R)= 0\) if the Ricci curvature of \(M\) is negative definite.