ΓΕ8 - Special Topics in Geometry
Syllabus
- Bochner’s technique in Differential Geometry.
- Complex manifolds, Kähler manifolds, Riemann surfaces.
- Isometric and conformal immersions.
- Rigidity aspects of isometric immersions.
- Minimal submanifolds in Riemannian manifolds.
- Harmonic maps, geometric PDE’s and flows.
ΓΕ1 - Differential Geometry
Syllabus
- Manifolds of the Euclidean space.
- Tangent and normal bundles.
- 1st and 2nd fundamental forms.
- Weingarten operator and Gauss map.
- Convex hypersurfaces.
- Hadamard’s Theorem.
- 1st and 2nd variation of area.
- Minimal submanifolds.
- Weierstrass representation.
- Bernstein’s Τheorem.
ΓΕ3 - Riemannian Geometry
Syllabus
- Riemannian metrics, isometries, conformal maps.
- Geodesics and exponential maps.
- Parallel transport and holonomy.
- Hopf-Rinow’s Theorem.
- Curvature operator, Ricci curvature, scalar curvature.
- Riemannian submanifolds.
- Gauss-Codazzi-Ricci equations.
- 1st and 2nd variation of length.
- Jacobi fields.
- Comparison theorems.
- Homeomorphic sphere theorem
ΓΕ2 - Differential Geometry
Syllabus
- Topological and smooth manifolds.
- Tangent and cotangent bundles.
- Vector fields and their flows.
- Submanifolds and Frobenius’ Theorem.
- Vector bundles.
- Connection and parallel transport.
- Differential forms.
- De Rham cohomology.
- Integration.
- Stokes’ Theorem.