Tensor algebra and tensor analysis were developed by Riemann, Christoffel, Ricci, Levi-Civita and others in the nineteenth century. The special theory of relativity, as propounded by Einstein in 1905, was elegantly expressed by Minkowski in terms of tensor fields in a flat space-time.
In 1915, Einstein formulated the general theory of relativity, in which the space-time manifold is curved. The theory is aesthetically and intellectually satisfying. The general theory of relativity involves tensor analysis in a pseudo-Riemannian manifold from the outset. Later, it was realized that even the pre-relativistic particle mechanics and continuum mechanics can be elegantly formulated in terms of tensor analysis in the three-dimensional Euclidean space. In recent decades, relativistic quantum field theories, gauge field theories, and various unified field theories have all used tensor algebra analysis exhaustively.
This book develops from abstract tensor algebra to tensor analysis in various differentiable manifolds in a mathematically rigorous and logically coherent manner. The material is intended mainly for students at the fourth-year and fifth-year university levels and is appropriate for students majoring in either mathematical physics or applied mathematics.
The first chapter deals with tensor algebra, or algebra of multilinear mappings in a general field F. (The background vector space need not possess an inner product or dot product.). The second chapter restricts the algebraic field to the set of real numbers R. Moreover, it is assumed that the underlying real vector space is endowed with an inner product (or dot product). Chapter 3 defines and investigates a differentiable manifold without imposing any other structure. Chapter 4 discusses tensor analysis in a general differentiable manifold. Differential forms are introduced and investigated. Next, a connection form indicating parallel transport is brought forward. As a logical consequence, the fourth-order curvature tensor is generated. Chapter 5 deals with Riemannian and pseudo-Riemannian manifolds. Tensor analysis, in terms of coordinate components as well as orthonormal components, is exhaustively investigated. In Chapter 6, special Riemannian and pseudo-Riemannian manifolds are studied. Flat manifolds, spaces of constant curvature, Einstein spaces, and conformally flat spaces are explored. Hypersurfaces and submanifolds embedded in higherdimensional manifolds are discussed in chapter 7. Extrinsic curvature tensors are defined in all cases. Moreover, Gauss and Codazzi-Mainardi equations are derived.
We would like to elaborate on the notation used in this book. The letters i, j, k, l, m, n, etc., are used for the subscripts and superscripts of a tensor field in the coordinate basis. However, we use the letters a, b, c, d, e, f, etc., for subscripts and superscripts of the same tensor field relative to an orthonormal basis. The numerical enumeration of coordinate components vi of a vector field is given by v1, v2, ... ,vN. However, numerical elaboration of orthonormal components of the same vector field is furnished by v(1), v(2), ..., v(n) (to avoid confusion). Similar distinctions are made for tensor field components. The flat metric components are denoted either by dij or dab. (The usual symbol η is reserved only for the totally antisymmetric pseudotensor of Levi-Civita.) The generalized Laplacian in the AT-dimension is denoted by Δ.
I would like to acknowledge my gratitude to several people for various reasons. During my stay at the Dublin Institute for Advanced Studies from 1958 to 1961, I learned a lot of classical tensor analysis from the late Professor J. L. Synge, F. R. S.. Professor W. Noll, a colleague of mine at Carnegie-Mellon University from 1963 to 1966, introduced me to the abstract tensor algebra, or the algebra of multilinear mappings. My research projects and teachings on general relativity for many years have consolidated the understanding of tensors. Dr. Andrew DeBenedict is has kindly read the proof, edited and helped with computer work. Mrs. Judy Borwein typed from chapter 1 to chapter 5 and edited the text diligently and flawlessly. Mrs. Sabine Lebhart typed the difficult chapter 7 and appendices. She also helped in the final editing. Mr. Robert Birtch drew thirty-four figures of the book. Last but not least, my wife, Mrs. Purabi Das, was a constant source of encouragement.