Riemannian and Hermitian Metrics

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

Chapter Chapter 7 in Encyclopedia of Distances, 2016, pp 135-168 from Springer

Abstract: Abstract Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of 2D 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 d s 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: 2016
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DOI: 10.1007/978-3-662-52844-0_7

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