Topics
in Advanced Geometry B

Introduction
to smooth manifolds

a.y. 2016/2017

We shall introduce the concept of an abstract n-dimensional
differentiable manifold. This is the natural space where the notion of
differentiability of a map can be introduced. The tangent bundle and
the subsequent theory of vector fields will be studied in detail. A
large part of the course will be devoted to the construction of
concrete examples of differentiable manifolds. This will be done by
means of different tools ranging from the implicit function theorem up
to smooth actions of discrete groups. We will see how algebraic
objects such as the linear group or the orthogonal group can be
endowed with a natural structure of a differentiable manifold.
Meanwhile, we shall present some concepts from the theory of
submanifolds and show, according to the (simplified version of a)
celebrated theorem by H. Whitney, that every compact abstract manifold
can be realized as a smooth subset of some Euclidean space of
sufficiently high dimension.

Schedule

- Tuesday, 2pm - 5pm, room 2.3 (Via Castelnuovo)
- Thursday,
10am - 1pm,
**room 2.3**(via Castelnuovo)