Sumários

Examples. Curvature of a curve at a point.

29 maio 2024, 11:30 • Sílvia Nogueira da Rocha Ravasco dos Anjos

Example of a calculation of the Gauss curvature and mean curvature of a surface of revolution in the Euclidean space, using the Gauss map. Minimal surfaces. 

Geometric interpretation of the sectional curvature. 
Normal curvature of a curve at a point. Principal curvatures. 
Curvature of a curve at a point. Examples. 

Isometric immersions

27 maio 2024, 12:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos

Killing-Hopf Theorem (continuation): examples of non-simply connected, geodesically completed manifolds with negative or positive constant curvature. 

Definition of isometric immersion f from N  to M. Second fundamental form of N. 

Example: if N is a hypersurface in M,  can define principal directions and principal curvatures at a point. Gauss curvature and mean curvature of f. If M is the Euclidean space with the Euclidean metric, can define the Gauss map. 

Relation between the sectional curvatures of N and M. 
Example: N is an hypersurface in M. If the dimension of N is 2 and M  is the Euclidean space equipped with the Euclidean metric then the Gauss curvature of N at each point is the Gauss curvature of f. 
(Gauss 's Theorema Egregium).  

Manifolds with constant curvature

22 maio 2024, 11:30 • Sílvia Nogueira da Rocha Ravasco dos Anjos

Shur Theorem:  A connected isotropic Riemnannian manifold of dimension greater or equal than 3 has constant curvature. 

Example: Hyperbolic space of radius a  has curvature -1/a^2. 

Actions of groups on topological manifolds: free and proper actions. 

Killing-Hopf Theorem: classification of  connected and geodetically complete Riemannian manifolds with constant curvature. 

Examples in dimension 2 with a flat metric. 

Gauss-Bonnet Theorem

20 maio 2024, 12:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos

The index of a vector field at a singular point is well defined: it does not depend on the choice of Riemannian metric, the choice of an orthonormal frame and on the choice of a neighbourhood  homeomorphic to a disc. Statement and proof of the Gauss-Bonnet Theorem. 

Corollaries: The integral of the Gauss curvature is the same for all Riemannian metrics. The sum of the indexes of a vector field with isolated singularities is the same for all vector fields on the manifold. It is called the Euler characteristic of the manifold. 
Triangulation of a surface. The Euler characteristic is the sum of number vertices minus the number of edges plus the number of faces of a triangulation. Examples. 
Manifolds with constant curvature: identify these manifolds using the curvature forms. 

Geodesic curvature. Index of a vector field at a singular point.

15 maio 2024, 11:30 • Sílvia Nogueira da Rocha Ravasco dos Anjos

Relation between the curvature form and the Gauss curvature of a surface. 

Dependence of the connection forms on the field of frames. Relation between the connection forms of two distinct field of orthonormal frames. 
Geodesic curvature: it is a measure of how much a curve fails to be a geodesic at each point. 
In particular, a curve is a geodesic iff its geodesic curvature vanishes. 
Singular point of a vector field. Isolated singularity. Definition of index of a vector field at a singular point. Examples.