The first eigenvalue of Laplace-type elliptic operators induced by conjugate connections

Abstract

According to Tashiro–Obata, on a Riemannian manifold (M, g) with its Ricci curvature bounded positively from below, the first eigenvalue of the Laplacian on functions satisfies a simple inequality in terms of the scalar curvature, and equality characterizes the Riemannian sphere. We discuss a similar inequality for a certain elliptic operator on a manifold with conjugate connections. As application we characterize hyperellipsoids in Blaschke’s unimodular-affine hypersurface theory.