Module page
Riemannian Geometry
| academic year: | 2013/2014 |
| instructor: | Andrea Sambusetti |
| degree course: | Mathematics (magistrale) |
| type of training activity: | caratterizzante |
| credits: | 6 (48 class hours) |
| scientific sector: | MAT/03 Geometria |
| teaching language: | italiano |
| period: | I sem (30/09/2013 - 17/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
- Preliminaries on submanifolds of the Euclidean space: tangent space, vector fields, differentiable functions and maps between submanifolds, (parametrizations, imemrsions, embeddings, submersions), directional derivatives of functions and differential of maps; tangent bundle, bracket, distributions, Frobenius' Theorem, tensors and differential forms; riemannian structure (I and II fundamental forms), shape operator, covariant derivative of submanifolds in E^n.
- Riemannian manifolds: examples; Levi-Civita connections and parallel transport; geodesics, exponential map, normal coordinates, conjugate and cut points; curvature, relation with the curvature of submanifolds of E^n (Gauss' Theorem); Jacobi fields, first and second variation formula.
- Some important theorems of Riemannian Geometry Cartan-Hadamard's Theorem, Synge's Theorem, first theorem of Cartan. Spazi forma e loro geometria.
- Space-forms and their geometry.
Suggested reading:
S. Gallot-D. Hulin-J. Lafontaine "Riemannian geometry", Graduate Texts in Math., Springer-Verlag, Berlin-Heidelberg-New York, 1993;
Do Carmo, "Riemannian Geometry", BIrkhauser, Boston, 1992;
A. Sambusetti "Complementi ed Esercizi di Geometria differenziale", http://www1.mat.uniroma1.it/people/sambusetti/andreas_webpage/geometria.html
Type of course: standard
Prerequisites:
- Linear Algebra. Elementary topology, fundamental group. Analysis in R^n: parametrizations and cartesian equations of subspaces of the Euclidean space (regularity, Dini's Theorem, Inverse Function Theorem, Lagrange multipliers etc). Knowledge of the notions of derivative and differentiability in R^n and on submanifolds of R^n. Knowledge of the basic theorems of integral calculus in R^n and on submanifolds of R^n (submanifolds with boundary, Green's and Stokes's theorems). Differential geometry of curves and surfaces in the Euclidean space.
Knowledge and understanding:
Successful students will be able to deal with topics concerning basic Riemannian geometry, including fundamental examples of Riemannian manifolds, and with its applications to astronomy, theory of relativity, theoretical physics, etc. They will become proficient and acquainted with subjects such as tensors, covariant derivatives, riemannian metrics.
Skills and attributes:
Successful students will be able to perform the principal operations of tensorial calculus, such as Lie derivation and differentiation of exterior forms. Compute the Lie algebra of a Lie group. Study of submanifolds of an euclidean space when they are definite by cartesian or parametric equations, both from the local and global point of view by computing the metric, the Levi Civita connection and the corresponding parallel transport, the geodesics, the curvature. Also, they will have some basic knowledge on how to use software Mathematica for study of differential geometry of curves and surfaces.
Personal study: the percentage of personal study required by this course is the 65% of the total.