GEOMETRY FOR TODAY

By Y. HIZAT

CONTENTS DIFFERENTIABLE MANIFOLDS TOPOLOGICAL MANIFOLDS DIFFERENTIABLE MAPS TANGENT SPACE IMMERSIONS AND EMBEDDINGS VECTOR FIELDS LIE GROUPS ORIENTABILITY MANIFOLDS WITH BOUNDARY NOTES SECTION 1 1 SECTION 1 2 SECTION 1 4 SECTION 1 5 SECTION 1 7 BIBLIOGRAPHICAL NOTES REFERENCES DIFFERENTIAL FORMS TENSORS TENSOR FIELDS INTEGRATION ON MANIFOLDS STOKES THEOREM ORIENTATION AND VOLUME FORMS SECTION 2 1 SECTION 2 4 SECTION 2 5 RIEMANNIAN MANIFOLDS AFFINE CONNECTIONS LEVI-CIVITA CONNECTION MINIMIZING PROPERTIES OF GEODESICS HOPF-RINOW THEOREM SECTION 3 5 CURVATURE CARTAN STRUCTURE EQUATIONS GAUSS–BONNET THEOREM MANIFOLDS OF CONSTANT CURVATURE ISOMETRIC IMMERSIONS SECTION 4 4 GEOMETRIC MECHANICS MECHANICAL SYSTEMS HOLONOMIC CONSTRAINTS RIGID BODY NON-HOLONOMIC CONSTRAINTS LAGRANGIAN MECHANICS HAMILTONIAN MECHANICS COMPLETELY INTEGRABLE SYSTEMS SYMMETRY AND REDUCTION SECTION 5 1 RELATIVITY GALILEO SPACETIME SPECIAL RELATIVITY THE CARTAN CONNECTION GENERAL RELATIVITY THE SCHWARZSCHILD SOLUTION COSMOLOGY CAUSALITY HAWKING SINGULARITY THEOREM PENROSE SINGULARITY THEOREM SOLUTIONS TO SELECTED EXERCISES CHAPTER 1 SECTION 1 3 SECTION 1 6 SECTION 1 8 SECTION 1 9 CHAPTER 2 SECTION 2 2 SECTION 2 3 SECTION 2 6 CHAPTER 3 SECTION 3 1 SECTION 3 2 SECTION 3 3 SECTION 3 4 CHAPTER 4 SECTION 4 1 SECTION 4 2 SECTION 4 3 SECTION 4 5 CHAPTER 5 SECTION 5 2 SECTION 5 3 SECTION 5 4 SECTION 5 5 SECTION 5 6 SECTION 5 7 SECTION 5 8 CHAPTER 6 SECTION 6 1 SECTION 6 2 SECTION 6 3 SECTION 6 4 SECTION 6 5 SECTION 6 6 SECTION 6 7 SECTION 6 8 SECTION 6 9 REFERENCE