Differential Geometry

• Describe and apply local charts (and diffeomorphisms between them) for Riemannian manifolds – in particular with the aim of representing the Riemannian metrics concretely as indicatrix fields in the chart domain.
• Describe, recognize, and apply isometries and conformal maps between Riemannian manifolds.
• Describe and apply tangent spaces, vector fields, considered as derivations, and their integral curves, considered as deformation maps (flow maps).
• Find and apply the Lie derivative, gradient, divergence, Hessian, and Laplace operators in Riemannian manifolds.
• Apply the concept of a tensor to analyze multilinear maps.
• Explain and apply the Levi-Civita connection in Riemannian manifolds.
• Explain and apply the notion of parallel transport on surfaces and in Riemannian manifolds.
• Determine geodesic curves and the exponential and logarithmic map on a surface and in a Riemannian manifold.
• Explain the construction and meaning of the curvature tensor, the Ricci curvature, the sectional curvature and the scalar curvature in Riemannian manifolds.
• Apply first and second variation of arc length to get global geometric and topological consequences of bounds on the curvature tensors.
• Apply simple extensions of the above concepts and results to Lorentzian and Finslerian manifolds.
• Apply differential geometric concepts and results to a wide spectrum of modern and significant modelling scenarios.

Diffeomorphisms; tangent spaces; metric tensors; Poincaré models; isometries; Lie derivative; Killing vector fields; Levi-Civita connection; covariant differentiation; parallel transport; geodesics; helices and circles in Riemannian 3-manifolds; fundamental differential operators; the Laplace equation; Exponential map; Logarithmic map; first and second variation of the arc length functional; geodesic balls and spheres and their volumes; curvature operator; curvature tensor; sectional curvature; Ricci curvature; scalar curvature; the Einstein tensor; applications to Newtonian mechanics, general relativity, and to Finslerian anisotropic geometric phenomena.

01125/01237, Knowledge about fundamental concepts and results from calculus, geometry, and linear algebra is assumed as a prerequisite.

Lectures and exercises.