Riemannian Geometry

  • Dieter LüstEmail author
  • Ward Vleeshouwers
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)


We consider (d+1)-dimensional smooth manifolds \(\mathcal {M}\), which are topological manifold that look locally like \(\mathbb {R}^n\). \(\mathcal {M}\) can be covered by open sets \(U_i\), \(i \in I\), where I is some indexing set. The charts are then defined as bijective maps \(\phi : U_i \rightarrow \mathbb {R}^{1,d}\) with the requirement that, for \(U_i \bigcap U_j \ne 0\), the transition function \(\phi _i \circ \phi _j^{-1}\) is \(C^{\infty }\). The collection of all \(U_i\) is then called an atlas.

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Arnold-Sommerfeld-CenterLudwig-Maximilians-UniversitaetMunichGermany
  2. 2.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands

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