Foundational Material

Abstract

In Section 1.1, we discuss the basic notions and results of vector spaces, vectors and tensors in them, and the general linear group. In Section 1.2, we consider the main topics associated with differentiable manifolds: tangent spaces, frame bundles, mappings, exterior differential calculus, Cartan’s lemma, completely integrable systems, the Frobenius theorem, Cartan’s test for a system in involution, the structure equations of a differentiable manifold and of the general linear group, and affine connections. Section 1.3 is dedicated to a projective space— we consider projective transformations, projective frames, and the structure equations of a projective space, the duality principle, projectivization, classical homogeneous spaces (affine, Euclidean, non-Euclidean), and their transformations. In Section 1.4 we demonstrate the geometric and analytic methods of specialization of moving frames by considering the geometry of a curve in the projective plane. Finally, in Section 1.5, we study some algebraic varieties, namely, Grassmannians and determinant submanifolds (Segre and Veronese varieties).KeywordsProjective SpaceProjective TransformationIntegral ManifoldDifferentiable ManifoldAffine ConnectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.