MATH 574 Modern Geometry II

Tensors. Algebraic Theory and transformation rules. Skew-symmetrical tensors. Differential forms. Tensors in Riemannian and pseudo-Riemannian spaces. Vector fields and Lie algebras. The Lie derivative. The fundamental matrix Lie algebras. The exterior derivative and integration of differential forms. The general Stokes formula. Differential forms on complex spaces. The Kahlerian metrics. The curvature form. Covariant differentiation and the metric. Parallel transport of vector fields. Geodesics. The Riemann curvature tensor. The general curvature tensor. The symmetries of the curvature tensor. Examples of the curvature tensor in spaces of dimensions 2 and 3. The simplest concepts of the general theory of relativity.