Instructor Prof. Sergey Finashin

1. Submanifolds in an Euclidian space (4 definitions).
2. Chart, atlas, differential structure. Diffeomorphisms. Raising of the class of a differential structure.
A line with an exotic differential structure. Exotic Milnor spheres. Simplicity of  Dif. Topology up to dimension 3.
3. Tangent and cotangent vectors. Tangent and cotangent vector bundles (with the dif. structure in the total spaces).
Vector fields and 1-forms. Differential of a function. Differential of a map between smooth manifolds. Tensor fields
and differential forms.
4. Manifolds with a boundary. Smoothing the corners in the product.
5. Submanifolds. Immersions, submersions, embeddings. Embedding and immersions of manifolds into an Euclidian space.
6. Orientation: comparing different definitions