Chapter 7 - Riemannian and Hermitian Metrics

Abstract

This chapter discusses about Riemannian geometry and Riemannian and Hermitian metrics. Riemannian geometry is a multidimensional generalization of the intrinsic geometry of two-dimensional surfaces in the Euclidean space E3. It focuses real smooth manifolds equipped with Riemannian metrics. Hermitian geometry focuses complex manifolds equipped with Hermitian metrics, that is, collections of positive-definite symmetric sesquilinear forms on their tangent spaces, which varies smoothly from point to point. It is a complex analog of Riemannian geometry. A special class of Hermitian metrics is Kahler metrics which have closed fundamental form w. A generalization of Hermitian metrics gives complex Finsler metrics which cannot be written in terms of a bilinear symmetric positive-definite sesqulinear form. The metric tensor (or basic tensor, fundamental tensor) is a symmetric tensor of rank 2, that is used to measure distances and angles in a real n-dimensional differentiable manifold Mn. A Riemannian metric on Mn is a collection of inner products <,>p on the tangent spaces Tp(Mn), one for each p element of Mn. A conformal structure on a vector space V is a class of pairwise-homothetic Euclidean metrics on V.