1. An Introduction to Riemannian Geometry
With Applications to Mechanics and Relativity
Presented by Mehdi Nadjafikhah ©
URL: webpages.iust.ac.ir/m_nadjfikhah

3. Contents
Chapter 1 - Differentiable Manifolds 1.1 Topological Manifolds 1.2 Differentiable Manifolds 1.3 Differentiable Maps 1.4 Tangent Space 1.5 Immersions and Embeddings 1.6 Vector Fields 1.7 Lie Groups 1.8 Orientability 1.9 Manifolds with Boundary

4. Contents
Chapter 2 - Differential Forms 2.1 Tensors 2.2 Tensor Fields 2.3 Differential Forms 2.4 Integration on Manifolds 2.5 Stokes Theorem 2.6 Orientation and Volume Forms

5. 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

6. Contents
Chapter 4 – Curvature 4.1 Curvature 4.2 Cartan Structure Equations 4.3 Gauss–Bonnet Theorem 4.4 Manifolds of Constant Curvature 4.5 Isometric Immersions

7. Contents
Chapter 5 - Geometric Mechanics 5.1 Mechanical Systems 5.2 Holonomic Constraints 5.3 Rigid Body 5.4 Non-holonomic Constraints 5.5 Lagrangian Mechanics 5.6 Hamiltonian Mechanics 5.7 Completely Integrable Systems 5.8 Symmetry and Reduction

8. Contents
Chapter 6 – Relativity 6.1 Galileo Spacetime 6.2 Special Relativity 6.3 The Cartan Connection 6.4 Relativity 6.5 Galileo Spacetime 6.6 Special Relativity 6.7 The Cartan Connection