Power Series for Volume of Balls in Riemannian Manifold

[tornado28]

I'm trying to work out the following problem: Find the first two terms of the power series expansion for the volume of a ball of radius r centered at p in a Riemannian Manifold, M with dimension n. We are given that

$$Vol(B_r(p)) = \int_S \int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t \mathrm{d}v$$

(I'm ignoring the cut locus distance because we are only interested in small r anyway.) I assume that $S = S^{n-1} = \{v \in T_pM : |v| = 1 \}$. Clearly the constant term is 0, because a ball of radius 0 has no area in any manifold, so I think that they are really asking for the first two non-zero terms. To find the next term we need to calculate the derivative of $Vol(B_r(p))$ wrt r.

$$\frac{\mathrm{d}}{\mathrm{d}r} Vol(B_r(p)) = \frac{\mathrm{d}}{\mathrm{d}r}\int_S \int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t \mathrm{d}v = \int_S \frac{\mathrm{d}}{\mathrm{d}r} \left(\int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t\right) \mathrm{d}v = \int_S \det(d(exp_p)_{rv})r^{n-1} \mathrm{d}v$$

I can pass the derivative under the integral sign because S is compact and the inner integral is a continuous function of v. (Right?) So the second term is 0 also, unless n=1. Here's where I run into trouble. I'm not sure how to differentiate $\det(\mathrm{d}(exp_p)_{rv})$. I do know that $\mathrm{d}(exp_p)_{tv} \cdot tw = J(t)$ where $J(t)$ is the Jacobi Field along $\gamma_v$ defined by $J(0)=0$ and $J'(0)=w$. My guess is that the solution uses this along with some linear algebra trick to find the derivative.

Thanks in advance for your help!

 

[Vargo]

My notation might be a bit rusty, but the differential of the exponential at 0 is the identity. So d(exp)(rv)= I + O(r).

So the first non zero term will come from that I, which has determinant 1. You integrate over v to get the usual surface area of a (Euclidean) sphere. Since you are looking at the derivative of V, the antiderivative gives the volume of the euclidean ball.

To get the second term, you'll need the next term in the differential of that exponential. It will have to involve curvature somehow. From your formula, it looks like you might take J(w,t)/t and find the second non zero term in its expansion. (The first non zero term is w, which is why d(exp)(0) = I.

 

[tornado28]

Thanks. That gets me the first term, and at least gives me an educated guess for the second term. I'm guessing that $\frac{d}{dr} \left( d(\exp_p)_{rv} \right) = ric_p(v)$. When you integrate this over the whole ball you end up getting the scalar curvature which is what was supposed to happen.

 

[Vargo]

Since you mentioned the Ricci tensor, I looked it up on Wikipedia. They have formulas there that look relevant. In particular, under the heading "direct geometric meaning", they have the first two nonzero terms in the Taylor expansion of the volume element in geodesic normal coordinates.

http://en.wikipedia.org/wiki/Ricci_curvature

 

[blue_raver22]

I would first like to commend you for tackling such a complex problem in Riemannian geometry. The power series expansion for the volume of a ball in a Riemannian manifold is a fundamental and challenging concept, and your approach so far is very thorough and well thought out.

To address your question about differentiating the determinant term, I would suggest using the fact that the derivative of a determinant is given by the trace of the adjoint matrix. In this case, the adjoint matrix would be the inverse of the matrix \mathrm{d}(exp_p)_{rv} multiplied by its cofactor matrix. This may require some additional calculations, but it should ultimately give you the desired result.

Additionally, I would like to mention that your assumption for S being the unit sphere in the tangent space at p is correct. This follows from the definition of a Riemannian manifold, where the metric is defined at each point and the tangent space at p is isomorphic to the tangent space at any other point on the manifold.

In conclusion, I believe that with some additional calculations and using the properties of determinants and adjoint matrices, you will be able to find the first two non-zero terms of the power series expansion for the volume of a ball in a Riemannian manifold. Good luck!