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.

Course Outline

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

Course Outline

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.

Course Outline

Syllabus

• Homology and cohomology.
• Betti numbers.
• Attaching and gluing manifolds.
• Morse functions.
• Sard’s Theorem.
• Passing through a critical value.
• Regular interval theorem.
• CW decomposition of manifolds.
• Morse inequalities.
• Total curvature and Gauss maps.

Course Outline