Mathematics

GABRIELE MONDELLO

LEARNING RESULT - ACQUIRED KNOWLEDGE:

-know the basic concepts of the differential geometry of the differentiable manifolds of R ^ n and of the differential calculus on them.
-know the basic results on curves and surfaces.
-know the notion of abstract Riemannian surfaces and manifolds, and the two fundamental theorems of Gauss and Gauss-Bonnet.
-know the basic elements of spherical and hyperbolic geometry.

At the end of the course the student will acquire basic knowledge of submanifold of Euclidean spaces and of differential topology.
Thanks to the assignment of exercise sheets, the student will be able to understand given proofs in differential topology and to determine their rigour, and will be to able to produce simple proofs in differential topology and to illustrate them with method and clarity, also replying to possible objections.
Moreover the student will achieve the ability to provide solutions to new problems and to evaluate such solutions with logic rigour.

Basic topology of R^n
Basic linear algebra
Differentiable multivariable calculus

Recall about the implicit function theorem
Differentiable submanifolds of Euclidean spaces
Differentiable functions and differentiable partitions of unity
Abstract differentiable manifolds and Whitney embedding theorem
Tubular neighbourhood theorem and Ehresmann fibration theorem
Sard's lemma, transversality, intersection
Orientation and orientability
Degree of a proper map
Vector fields, integral curves, flows
Index of an isolated zero of a vector field
Poincaré-Hopf theorem

J. Milnor, "Topology from the differentiable viewpoint", Princeton landmarks in Mathematics
V. Guillemin, A. Pollack, "Differential topology", American Mathematical Society.

Other possibly useful texts:
J. Lee, "Introduction to smooth manifolds" (second edition), Graduate Texts in Mathematics, Springer
M. Manetti, Dispense di geometria differenziale (2008)

In-person lectures in class (66%).
In-person recitations in class (34%).

Optional but recommended. In person.

The exam consists of a written test (in which the student will be asked to solve problems of the same type as those carried out in the exercises) and an oral exam (in which the student will be asked questions about the topics illustrated during the course). The written test will last about two hours.
In order to get the passing grade 18/30, the student must show basic knowledge of the main topics, and to be able to carry out at least the simplest exercises among the assigned ones.
To achieve a score of 30/30 cum laude, the student must show an excellent knowledge of all the topics covered during the course and to be able to connect them in a logical and consistent way.

Determine whether the following subset of R^n is a submanifold, and whether it is compact or connected.
Determine whether the following map is proper: if so, compute its degree.
Determine the locus of critical points and of critical values of the following map.
Determine the isolated critical points of the following vector field and compute their index.
Determine whether the following submanifolds are transverse to each other.