1. An Introduction to Riemannian Geometry Chapter 3 Presented by Mehdi Nadjafikhah © URL: webpages.iust.ac.ir/m_nadjfikhah Email: m_Nadjafikhah@iust.ac.ir 5 April 2016 17 September 1826 – 20 July 1866

2. Contents Chapter 3 - Riemannian Manifolds 3.1 Riemannian Manifolds 3.2 Affine Connections 3.3 Levi-Civita Connection 3.4 Minimizing Properties of Geodesics 3.5 Hopf-Rinow Theorem

3.  The metric properties of n-dimensional Euclidean space (distances and angles) are determined by the canonical Cartesian coordinates.  In a general differentiable manifold, however, there are no such preferred coordinates;  to define distances and angles one must add more structure by choosing a special 2-tensor field, called a Riemannian metric.  This idea was introduced by Riemann in his 1854 habilitation lecture “On the hypotheses which underlie geometry”,  following the discovery (around 1830) of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky.  It proved to be an extremely fruitful concept, having led, among other things, to the development of Einstein’s general theory of relativity.

4. http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf

5. 3.1 Riemannian Manifolds

9. Example 1.5

12. Example: Helicoid

17. The Tangent–Cotangent Isomorphism

20. Inverse metric on Polar plane

21. Killing vector field

22. Killing vectors of the Poincare half-plane Inverse metric on Polar plane

23. 3.2 Affine Connections

25. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space R n by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.Euclidean spaceaffine space É lie Joseph Cartan 1869-1951

26. Elwin Bruno Christoffel 1829-1900

28. Remark 2.3  Locally, an affine connection is uniquely determined by specifying its Christoffel symbols on a coordinate neighborhood.  However, the choices of Christoffel symbols on different charts are not independent, as the covariant derivative must agree on the overlap.

30. Geodesic

32. Parallel transportaion of vector field Y along curve C with respect to the connection Gamma.

33. General parallel vector field Y along curve C.

35. Covariant Derivatives of Tensor Fields

36. This connection satisfies the following additional properties:

41. 3.3 Levi-Civita Connection Tullio Levi-Civita 1873-1941

42. Jean-Louis Koszul French Mathematician 1921-

48. Example: Sphere

50. Example: Hyperbolic Spaces

53. Divergence

55. Laplacian Laplacian of a function f defined as

56. 3.4 Minimizing Properties of Geodesics

59. Normal ball and Normal sphere

61. Lemma. X is (and hence the geodesics through p are) orthogonal to normal spheres. Lemma. X is (and hence the geodesics through p are) orthogonal to normal spheres.

73. 3.5 Hopf-Rinow Theorem

74. Geodesically complete