Riemannian and Hermitian Metrics

  • Michel Marie Deza
  • Elena Deza


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.


Riemannian Manifold Vector Bundle Tangent Space Complex Manifold Tangent Bundle 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michel Marie Deza
    • 1
  • Elena Deza
    • 2
  1. 1.École Normale SupérieureParisFrance
  2. 2.Moscow State Pedagogical UniversityMoscowRussia

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