nLab Riemannian geometry - contents

Riemannian geometry

Basic definitions

• Riemannian manifold

  • metric, isometry, isometry group

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

  • Lorentzian manifold, spacetime

• geodesic

  • geodesic convexity

  • geodesic flow

• Levi-Civita connection

  • Riemann curvature, torsion of a metric connection

Further concepts

• Hodge inner product, Hodge star operator

• gradient, gradient flow

Theorems

• Poincaré conjecture-theorem

Applications

• gravity

  • Einstein-Hilbert action, Einstein equations, supergravity

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