Differential Geometry [SMATB214]

Learning outcomes

This lecture is an introduction to the modern tools of tensor calculus on differentiable manifolds and riemanian geometry. Numerous geometrical and algebraic objects such as vectors, forms, tensors and related fields, curves and surfaces are introduced and generalised to non-euclidean geometry in an abstract way. The course also introduces students to tensor calculus and its techniques that are at the basis of numerous fields in fundamental and applied mathematics, or theoretical physics.

Content

The course presents the abstract generalisation of many objects of geometry, linear algebra and differential calculus. Riemanian differential geometry is presented as a smart and powerful synthesis of elementary algebra, geometry and analysis. The course is divided into four parts: (i) differentiable manifolds and tensor calculus (tensors and tensor fields , curves, surfaces and sub-manifolds, diffeomorphisms, flows and Lie derivatives) ; (ii) curvature and torsion on differentiable manifolds (parallel transport, affine connection, covariant derivative, curvature and torsion) ; (iii) riemanian and pseudo-riemanian geometry (metric and pseudo-metric, geodesics, Levi-Civita connection, Riemann tensor and applications) and (iv) diffenrential forms and exterior calculus (exterior derivative and exterior algebra, volume form, geometrisation of vector calculus, integration of differential forms and Cartan structure equations).