What is Differential Geometry?

Before we get into differential geometry, let’s talk a bit about some of the other little geometric things we’ve played with in the past.

First, let’s talk about the dot product. As you’ll recall, if we have vectors a=(a_1,a_2,...,a_n),b= (b_1,b_2,...,b_n) \in \mathbb{R}^n, the dot product of a and b, denoted by a \cdot b, is defined as

a \cdot b = a_1b_1 + a_2b_2 + \cdots + a_nb_n.

Now, it turns out that the dot product can be used to define all kinds of geometric objects. Here’s a list of some of the things that we can define in terms of the dot product:

  1. Length of a vector: \lvert a \rvert = \sqrt{a \cdot a}
  2. Angle between two vectors: the value \theta \in [0, \pi] such that \cos\theta = \frac{a \cdot b}{\lvert a \rvert \lvert b \rvert}
  3. Notion of orthogonality: we say that a and b are orthogonal if a \cdot b = 0
  4. Length of a curve: if \gamma : [a,b] \rightarrow \mathbb{R}^n is a smooth curve, the length of \gamma is \int\limits_a^b = \lvert \gamma'(t) \rvert dt.

In other words, the dot product can be used to define lots of geometric ideas.

Of course, we have other useful tools for measuring geometric quantities, including the integral (which is useful for measuring areas, volumes, that sort of thing) and the derivative (which is useful for measuring how curved or bent things are).

My point is that, when it comes to measuring geometric properties, we are big fans of the dot product, derivative, and integral.

These things all come naturally in $\mathbb{R}^n$, which is great news if you want to do all of your geometry in that setting. Of course, there are many other places one might want to study geometry. What about, say, topological manifolds?

And that’s exactly the idea. Differential geometry is about putting these objects onto topological manifolds, allowing us to study their geometry. The general technique isn’t too hard to conceptualize, although the formal mechanisms that support it take a bit of development.

The idea goes something like this: given a set M, we associate with M a set of maps that we will call our atlas. Formally, for some fixed n, the atlas satisfies this:

  1. For every p \in M, there is an open set $U_p \subseteq M$ and another open set $V \subseteq \mathbb{R}^n$, and a continuous map $\varphi : U_p \rightarrow V$ which has a continuous invers.
  2. For every pair of maps \varphi : V \rightarrow U, \psi : S \rightarrow T, if V \cap S \neq \emptyset, we ask that the map \varphi \circ \psi^{-1} : (\psi(V \cap S) \subseteq \mathbb{R}^n) \rightarrow (U \subseteq \mathbb{R}^n) have continuous mixed partial derivatives of all orders.
  3. There does not exist a map satisfying (1) and (2) that is not in the atlas (in this sense, the atlas is “maximal,” in that it is the “largest” set of maps satisfying (1) and (2)).

Once we have an atlas for our set, we can use that atlas to “import” ideas from \mathbb{R}^n into our set M. A set M together with the type of atlas defined above is called a smooth n-manifold.

For example, suppose you have a map f : M \rightarrow M. We say that this map is smooth if \phi \circ f \circ \varphi^{-1} has continuous mixed partial derivatives of all orders for all maps \phi,\varphi in the atlas of M (on the domain where this composition is defined).

See what we did there? By cleverly using our atlas, we took a map that had nothing to do with \mathbb{R}^n and trasported it into \mathbb{R}^n, allowing us to subject it to various tools from multivariable calculus. This is a theme in the development of smooth manifold theory, which is the framework upon which differential geometry is constructed.

So what are some of the interesting things that happen in differential geometry? Here’s a partial list:

  1. Tangent spaces: Think of a 2-d surface embedded in \mathbb{R}^3. For each point on this surface, if the surface is smooth at that point, we can talk about the “plane tangent to the surface at this point.” One important thing to remember: the tangent space always has the same dimension as the surface to which it is tangent.
    Given a smooth manifold M, and a point p \in M, we want to talk about the space that is tangent to M at p. Unfortunately, since we do not assume that M is embedded in some ambient space, say, \mathbb{R}^{n+1}, there is not an ambient space for us to extract our “tangent space” from.
    So here’s what we do: we define a new vector space that happens to be isomorphic to $\latex \mathbb{R}^n$, and whose elements are defined in terms of our point p \in M. The really cool thing about this space is that its elements are actually made up of functions (called derivations) that act on functions defined on the manifold at that point. Intuitively, they serve to generalize the notion of the “directional derivative.” It’s really wild.
  2. Coordinate-independence: There are lots of times when you’re working in a vector space and your answer depends on your choice of basis. One big theme in smooth manifold theory is coordinate independent representations. In other words, the results of your computations are usually invariant under change of coordinates.
  3. Tensors: Given a vector space V, a covariant k-tensor is a map T : {V \times \cdots \times V}_{k \text{copies}} \rightarrow \mathbb{R} that acts linearly in each of its arguments. Example: an inner product in a real Hilbert space is a covariant 2-tensor.
    As stated above, at each point p \in M, we defined a new n-dimensional vector space called the tangent space at that point. A ktensor field is a function that, given a point on the manifold, gives you a k-tensor defined on the tangent space at that point.
    These things are way cool. You can use them to define Riemannian metrics (which is the entry point to Riemannian geometry), and you can use them to define the de Rham cohomology (a useful tool for studying your manifold’s topology).

That’s just some of the cool stuff that’s going on there. If you want to get into this some more, I suggest reading Introduction to Smooth Manifolds by John Lee.

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