During the travelling time on train, I review the content of differential geometry - a topic that I have learned many times and long time ago, but handle so little in hand. I believe that I have learned it for at least three times : first time in Hong Kong, second time at Fudan and third time at Polytechnique. Some times later, I finally get understood its main idea, so in this short blog, I attempt to give a very non-rigorous introduction.
The first step is sometimes the most difficult step : to well define the object of manifold $\mathcal{M}$. Generally speaking, it is something similar to $\mathbb{R}^d$. But how it looks like? The standard definition of a manifold uses the language of local coordinate and atlas, I have to say that a very visualization is to imagine that a RPG guy runs in a game, and local we have to draw the map as a surface. We require that the $C^k$ condition to make locally the bijection is OK and we can always to approximation properly. Then, the compatibility condition makes all the local map together and define a good manifold.
The mapping between the manifold is used to define the equivalent class. The notation is complex but the idea is simple : when we talking about the property of squares, circles etc, we never identify a specific square of circle since they have the same property. For a manifold, we should also say the same story. That's why we invent this idea of the mapping between different manifold.
For $p \in \mathcal{M}$, its local increment has $d$ directions induced by the local map, so it has naturally a tangent plane $T_p\mathcal{M}$. When we remove the point $p$, it becomes a tangent fiber $T\mathcal{M}$. But until now, we say nothing about the distance and volume on the manifold. We have to add a metric $g$ to the manifold $(\mathcal{M}, g)$, which is a symmetric matrix, or a $(0,2)$-tensor. $g = g_{i,j}dx^i \otimes dx^j$. Then the curves can be written as
$$
L = \int_a^b \sqrt{g(\frac{d\gamma}{dt}, \frac{d\gamma}{dt})} dt,
$$
and the volume can be written as
$$
\int f d\mathcal{M} = \int_{\mathbb{R}^d} f \sqrt{det(g)} dx_1 dx_2 \cdots dx_d.
$$
We can check that all these definitions do not depend on the choice of local coordinate.
All these definitions make the manifold similar to $\mathbb{R}^d$ except some more profound for curvature. It starts from vector field $X$, which is generally not commutative, so we have the Lie crochet $[X, Y] = XY - YX$. And some more complicated idea is the covariant derivative (connection) written as $\nabla_{X}Y$. They develop the idea for curvature, Ricci curvature and scalar curvature. I skip these part since they may be more useful for the expert of geometry.
In the last par of the lecture book, people use these language to study the theory of relativity. For example, main idea is to find possible $g$ that makes the physical model work by some minimum action. Interested readers can check the lecture note of MAT568 by Jeremie Szeftel.
