Riemannian and Hermitian Metrics

Michel Marie Deza and Elena Deza
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Michel Marie Deza: École Normale Supérieure
Elena Deza: Moscow State Pedagogical University

Chapter Chapter 7 in Encyclopedia of Distances, 2013, pp 125-155 from Springer

Abstract: Abstract Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of two-dimensional surfaces in the Euclidean space $\mathbb{E}^{3}$ . It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point to point. The geometry of such (Riemannian) manifolds is based on the line element ds 2=∑ i,j g ij dx i dx j . This gives, in particular, local notions of angle, length of curve, and volume.

Keywords: Riemannian Manifold; Vector Bundle; Tangent Space; Complex Manifold; Tangent Bundle (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-3-642-30958-8_7

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