Projective Space in Differential Geometry

Question

I'm currently reading M. do Carmo's, "Riemannian Geometry." I've noticed that projective $n$-space $\mathbb{RP}^n$ or $\mathbb{CP}^n$ is often used as an example in this differential geometry text and several others. As far as I can tell, projective space is mostly applicable to algebraic geometry. Why is projective space given as an example in differential geometry texts, when it doesn't seem to be much used in this area?

Answer 1

Well, I think one of the reasons is that it is one of the first non trivial examples of $n$-manifold, so an Hausdorff and II countable topological space which admits at each point an open neighbourhood homeomorphic to $\mathbb R^n$ (resp. $\mathbb C^n$) and such that the transition functions are smooth (resp. holomorphic).

When you are going to study objects and maps between these objects (so basically a Category), first of all we need to give examples (possibly non trivial ones) where we can start to develop our theory.

Usually, once one starts the beautiful theory of smooth manifolds, the first examples are the trivial spaces $U\subseteq \mathbb R^n$, with $U$ open set of $\mathbb R^n$, then the $n$-sphere $S^n:=\lbrace x\colon \vert\vert x\vert\vert=1\rbrace\subseteq \mathbb R^n$. Here, the local charts are given for example via the stereographic projections. Later on, one usually talks about the projective space $\mathbb P^n$, whose local charts are the following

$$\phi_i\colon \lbrace x_i\neq 0\rbrace \to \mathbb R^n, \qquad \left(x_0,\dots, x_n\right)\mapsto (\frac{x_0}{x_i}, \dots, \frac{x_n}{x_i}).$$

However, I think there are several other reasons. In general, it depends by what are you going to study or to do in the theory of manifolds and varieties.

Another interesting fact is that usually if you have a conjecture or theory that needs to establish if it is true or not, then a projective space (or its sub-varieties) are good candidate to test it, basically because it is easy to take in exam (since it is easy to menage its charts a transitions functions).

Answer 2

The Fubini-Study metric on $\mathbb{CP}^n$ is arguably the simplest Riemannian metric after the constant curvature ones on $S^n$, $\mathbb{R}^n$, $\mathbb{H}^n$ and their discrete quotients. In particular, if you want to test a conjecture on a non-constant-curvature space, then $\mathbb{CP}^n$ is a good candidate.

As the "next simplest" metric after the constant-curvature ones, the Fubini-Study metric has lots of nice properties --- it is Einstein, it has strictly positive sectional curvature, and it is symmetric, for example --- and shows up in many classification theorems in both Riemannian and Kahler geometry.

Answer 3

One other reason one might wish to introduce projective spaces as examples of manifolds is that they are one of the first examples that one encounters without an explicit description as a subspace of $\Bbb{R}^n$. Indeed, $\Bbb{RP}^n$ (resp. $\Bbb{CP}^n$) is defined as the set of lines $\Bbb{R}^{n+1}$ (resp. $\Bbb{C}^{n+1})$ equipped with a topology and charts etc.

These are spaces that are abstract manifolds, but that are not embedded manifolds in $\Bbb{R}^n$ from their definition. Of course, in light of Whitney's embedding theorem this is true a posteriori, but the fact remains that these spaces are best initially described as abstract manifolds.