Titlebook: Introduction to Riemannian Manifolds; John M. Lee Textbook 2018Latest edition Springer Nature Switzerland AG 2018 Riemannian geometry.curv

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Model Riemannian Manifolds,folds” that should help to motivate the general theory. These manifolds are distinguished by having a high degree of symmetry. We begin by describing the most symmetric model spaces of all—Euclidean spaces, spheres, and hyperbolic spaces. Then we explore some more general classes of Riemannian manifolds with symmetry.

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,The Gauss–Bonnet Theorem,ms in Riemannian geometry, it asserts the equality of two very differently defined quantities on a compact Riemannian 2-manifold: the integral of the Gaussian curvature, which is determined by the local geometry, and . times the Euler characteristic, which is a global topological invariant.

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978-3-030-80106-9Springer Nature Switzerland AG 2018

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Introduction to Riemannian Manifolds978-3-319-91755-9Series ISSN 0072-5285 Series E-ISSN 2197-5612

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Connections,Before defining a notion of curvature that makes sense on arbitrary Riemannian manifolds, we need to study ., the generalizations to Riemannian manifolds of straight lines in Euclidean space. In this chapter, we introduce a new geometric construction called a ., which is an essential tool for defining geodesics.