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[CE] Cheeger and Ebin: Comparison Theorems in Riemannian Geometry

(一)

Chen-Yun continued reporting her thesis in theinformal general relativity seminar this Friday. The first problem is dealing with a nonlinear parabolic equation. The lesson I learned is we have to be very familiar with the linear theory and very careful when we encounter a nonlinear equation. There are questions I never thought of when I read GT. For example, in Schauder estimate for $a^{ij}D_{ij} u + c u = f,$ does the final constant $C$ depend linearly on $c$? Could you answer this in 1 minute? I supposed to do that in the seminar. After all, I am preparing my orals. I think the answer is no. (That means Chen-Yun may have a big trouble.)

Reading GR 4.5. One interesting question is, after we construct the generalized Newtonian potential $w,$ how do you show $u-w$ is harmonic in the weak sense, $\int_\Omega D_i  w D_i \phi = \int_\Omega f^i D_i \phi$ for any $\phi \in C^\infty_0(\Omega)?$ I tried to use the the crazy formula of $D_i w$ but fails after one or two hours. I finally asked Savin and he gave me a two-line argument.

(二)

I read the section "Curvature determines metric" in dC. This is the local version of Cartan-Ambrose-Hicks theorem in CE, which I don't understand their argument. It's a good theorem helping you use the Jacobi fields. Why couldn't I remember the idea now?

I did another exercise settled by myself: showing the metric on $\mathds{C}P^n$ coming from the submersion of $S^{2n+1} \rightarrow \mathds{C}P^n$ is the Fubini-Study metric we see in complex geometry. I think I get the answer except the final transition between the Riemannian metric and Hermitian metric that I never fully understand. That reminds me the sweet struggle with xxx in our complex geometry seminar.

Another beautiful section in GHL: 3.58-use Jacobi field on $\mathds{C}P^n$ to calculate the sectional curvature. The starting point is we have an explicit description of a family of geodesics on $S^{2n+1}.$ So we can write down the variational field and do the computation. GHL gives the result of two main cases but it still took me a long time figuring out how to get the formula of the general case. For the previous two, we can get the curvature on every points on the geodesic. For the general case, however, I can only get that at one point(It's similar to calculating in normal coordinates). This may be the difficulty for me.

As always, I recommend  this section.