Tensors

Tensors and tensor algebra are presented. The concept of a tensor is defined in two ways: as something which yields a scalar from a set of vectors, and as something whose components transform a given way. The meaning and use of these definitions is expounded carefully, along with examples. The action of the metric and its inverse (index lowering and raising) is derived. The relation between geodesic coordinates and Christoffel symbols is obtained. The difference between partial differentiation and covariant differentiation is explained at length. The tensor density and Hodge dual are briefly introduced.

WEYL SPACES OF COMPOSITIONS OF SIX BASIC MANIFOLDS

By help of prolonged covariant differentiation, Cartesian compositions of six basic manifolds are studied. Weyl spaces of such compositions are characterized. Eleven-dimensional Riemannian spaces containing compositions of six basic manifolds are also considered.

The Lanczos potential and Chern–Simons theory

In electromagnetism, the Faraday tensor [Formula: see text] can be constructed from the vector potential [Formula: see text], it is possible to add term to the Lagrangian depending on [Formula: see text] but not its derivatives called Chern–Simons terms. In gravitation, the Weyl tensor [Formula: see text] can be constructed from the Lanczos potential [Formula: see text], I pursue the analogy to see if terms of Chern–Simons form can be added to the Lagrangian. A new tensor [Formula: see text] is introduced which is constructed from the Lanczos potential and is of the same form as that of the Weyl tensor [Formula: see text] expressed in terms of the Lanczos potential except that covariant differentiation is replaced by transvection with a vector [Formula: see text]. The new tensor has associated invariants [Formula: see text] and [Formula: see text], the first of these can be interpreted as a Chern–Simons term for Weyl [Formula: see text] gravity. Both invariants allow various tensors to be constructed and some of their properties are investigated by using exact examples.

An Alternative Approach to Lie Groups and Geometric Structures

This book is about the foundations of geometric symmetry, namely, Lie groups and differential geometry. Although this is a classical subject about which hundreds of books have been written, this book takes a new and innovative approach. The main idea is to replace the Maurer–Cartan form with absolute parallelism and its curvature. Unlike the classical approach, where the model is fixed beforehand by the Maurer–Cartan form, this new approach is model-free, and also revisits the foundational concepts of differential geometry, such as covariant differentiation, from a different perspective.