Basic Riemannian Geometry

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  Bennet Chow       Peng Lu       Lei Ni

  1. Introduction
  2. Basic conventions(常規) and formulas in Riemannian geometry
  3. Laplacian and Hessian comparison theorems
  4. Geodesic polar coordinates
  5.  First and second variations of arc length and energy formulas
  6. Geometric application of second variation
  7. Green   function
  8. Comparison theory for the heat kernel
  9. Parametrix for the heat equation
  10. Eigenvalues and eigenfunctions of the Laplacian
  11. The determinant of the Laplacian
  12. Monotonicity for harmonic functions and maps
  13. Lie groups and left invariant metrics
  14. Bieberbach Theorem
  15. Compendium of inequalities
  16. Notes and commentary