Introdue to the study of Riemannian geometry, in particular by addressing the theorems of Gauss-Bonnet and Hopf-Rinow.
Curriculum
teacher profile teaching materials
Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using the study of geodesics; namely
the curves which, at least locally, minimize the distance on a Riemannian manifold.
2. Integration on surfaces. Area of a surface and total Gaussian curvature.
3. Covariant derivative. Covariant derivative of a tangent vector field. Parallel transport and
geodesics. Geodesic curvature.
4. Gauss-Bonnet theorem. Proof of Gauss-Bonnet theorem, local and global version. Relations between topology and geometry of surfaces.
5. Hopf-Rinow theorem. Riemnnian manifolds of arbitrary dimension. The exponential map in Riemannian geometry. Convex neighborhoods. Complete
manifolds: proof of Hopf-Rinow theorem. Applications: rigidity of the sphere.
6. Exercises. Written exercises at home and in class.
[1] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976). [2] M. Do Carmo , Riemannian Geometry. Birk ̈auser, (1992).
[3] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
[4] Marco Abate, Francesca Tovena, Geometria Differenziale. Springer, (2011).
Mutuazione: 20410444 GE430 - GEOMETRIA RIEMANNIANA in Matematica LM-40 PONTECORVO MASSIMILIANO
Programme
1. Introduction. We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim is to proveGauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using the study of geodesics; namely
the curves which, at least locally, minimize the distance on a Riemannian manifold.
2. Integration on surfaces. Area of a surface and total Gaussian curvature.
3. Covariant derivative. Covariant derivative of a tangent vector field. Parallel transport and
geodesics. Geodesic curvature.
4. Gauss-Bonnet theorem. Proof of Gauss-Bonnet theorem, local and global version. Relations between topology and geometry of surfaces.
5. Hopf-Rinow theorem. Riemnnian manifolds of arbitrary dimension. The exponential map in Riemannian geometry. Convex neighborhoods. Complete
manifolds: proof of Hopf-Rinow theorem. Applications: rigidity of the sphere.
6. Exercises. Written exercises at home and in class.
Core Documentation
[1] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976). [2] M. Do Carmo , Riemannian Geometry. Birk ̈auser, (1992).
[3] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
[4] Marco Abate, Francesca Tovena, Geometria Differenziale. Springer, (2011).
Type of delivery of the course
Frontal lecturesType of evaluation
The final grade will be based upon homework and written examination teacher profile teaching materials
Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using the study of geodesics; namely
the curves which, at least locally, minimize the distance on a Riemannian manifold.
2. Integration on surfaces. Area of a surface and total Gaussian curvature.
3. Covariant derivative. Covariant derivative of a tangent vector field. Parallel transport and
geodesics. Geodesic curvature.
4. Gauss-Bonnet theorem. Proof of Gauss-Bonnet theorem, local and global version. Relations between topology and geometry of surfaces.
5. Hopf-Rinow theorem. Riemnnian manifolds of arbitrary dimension. The exponential map in Riemannian geometry. Convex neighborhoods. Complete
manifolds: proof of Hopf-Rinow theorem. Applications: rigidity of the sphere.
6. Exercises. Written exercises at home and in class.
[1] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976). [2] M. Do Carmo , Riemannian Geometry. Birk ̈auser, (1992).
[3] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
[4] Marco Abate, Francesca Tovena, Geometria Differenziale. Springer, (2011).
Mutuazione: 20410444 GE430 - GEOMETRIA RIEMANNIANA in Matematica LM-40 PONTECORVO MASSIMILIANO
Programme
1. Introduction. We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim is to proveGauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using the study of geodesics; namely
the curves which, at least locally, minimize the distance on a Riemannian manifold.
2. Integration on surfaces. Area of a surface and total Gaussian curvature.
3. Covariant derivative. Covariant derivative of a tangent vector field. Parallel transport and
geodesics. Geodesic curvature.
4. Gauss-Bonnet theorem. Proof of Gauss-Bonnet theorem, local and global version. Relations between topology and geometry of surfaces.
5. Hopf-Rinow theorem. Riemnnian manifolds of arbitrary dimension. The exponential map in Riemannian geometry. Convex neighborhoods. Complete
manifolds: proof of Hopf-Rinow theorem. Applications: rigidity of the sphere.
6. Exercises. Written exercises at home and in class.
Core Documentation
[1] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976). [2] M. Do Carmo , Riemannian Geometry. Birk ̈auser, (1992).
[3] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
[4] Marco Abate, Francesca Tovena, Geometria Differenziale. Springer, (2011).
Type of delivery of the course
Frontal lecturesType of evaluation
The final grade will be based upon homework and written examination