Looking for a topic on differentiable manifolds

Question

I just finished a course on differentiable manifolds covering:
Definition of differentiable manifold.

Tangent space.
Submanifold.
Partitions of unity.
Vector fields.

And I looking for a topic for a half an hour presentation (with the knowledge I have so far ) on manifolds.
The half hour restriction is the real problem here, because all the topics I find can’t be presented in such sort time.
An example of a topic ( which is unfortunately taken is the exponential map ).
Any ideas?
( I have to present a topic to show my professor I have a good understanding of the courses material)

P.S \ is an introduction to complex manifolds possible in this time period?

Answer 1

You can go on to talk about covector fields then general tensor fields both in local coordinates or in an invariant fashion. I recommend at least doing the local coordinates approach to get a "hands on" feel for the subject. You could then define the Riemannian metric tensor as a special $(0,2)$-tensor. If you still have time, you can also define the Levi-Civita connection, Lie bracket of vector fields, and even define the Riemann curvature tensor.

Depending on your speed, a subset of the above should constitute a great 30-minute talk.