Table of Contents

1. Tensors and their Algebra 1.1 Introduction 1.2 Transformation of Coordinates 1.3 Summation Convention 1.4 Kronecker Delta 1.5 Scalars, Contravariant and Covariant Vectors 1.6 Tensors of Higher Rank 1.7 Symmetry of Tensors 1.8 Algebra of Tensors, Addition and Subtraction, Equality of Tensors, Inner and Outer Products, Contraction, The Quotient Law 1.9 Irreducible Tensors, Exercises 2. Riemannian Space and Metric Tensor 2.1 Introduction 2.2 The Metric Tensor 2.3 Raising and Lowering of Indices-Associated Tensor 2.4 Vector Magnitude 2.5 Relative and Absolute Tensors 2.6 The Levi-Civita Tensor, Exercises 3. Christoffel Symbols and Covariant Differentiation 3.1 Introduction 3.2 Christoffel Symbols 3.3 Transformation Law for Christoffel Symbols 3.4 Equation of a Geodesic 3.5 Affine Parameter 3.6 Geodesic Coordinate System 3.7 Covariant Differentiation, Covariant Derivatives of Contravariant and Covariant Vectors, Covariant Derivatives of Rank Two Tensors, Covariant Derivatives of Tensors of Higher Rank 3.8 Rules for Covariant Differentiation 3.9 Some Useful Formulas, Divergence of a Vector Field, Gradient of a Scalar and Laplacian, Curl of a Vector Field, Divergence of a Tensor Field 3.10 Intrinsic Derivative-Parallel Transport 3.11 Null Geodesics 3.12 Alternative Derivation of Equation of Geodesic, Exercises 4. The Riemann Curvature Tensor 4.1 Introduction 4.2 The Riemann Curvature Tensor 4.3 Commutation of Covariant Derivatives 4.4 Covariant Form of the Riemann Curvature Tensor 4.5 Properties of the Riemann Curvature Tensor 4.6 Uniqueness of the Riemann Curvature Tensor 4.7 The Number of Algebraically Independent Components of the Riemann Curvature Tensor 4.8 The Ricci Tensor and the Scalar Curvature 4.9 The Einstein Tensor 4.10 The Integrability of the Riemann Tensor and the Flatness of the Space 4.11 The Einstein Spaces 4.12 Curvature of a Riemannian Space 4.13 Spaces of Constant Curvature, Exercises 5. Some Advanced Topics 5.1 Introduction, 5.2 Gewodesic Deviation 5.3 Decomposition of Riemann Curvature Tensor 5.4 Electric and Magnetic Parts of the Riemann and Weyl Tensors 5.5 Classification of Gravitational Fields 5.6 Invariants of the Riemann Curvature Tensor 5.7 Curvature Tensors Involving the Riemann Tensor, Space-matter Tensor, Conharmonic Curvature Tensor 5.8 Lie Derivative 5.9 The Killing Equation 5.10 The Curvature Tensor and Killing Vector, Exercises

About the Author

ZAFAR AHSANDept of Mathematics, Aligarh University, India