The question

Let $(M,g)$ be a Riemannian manifold of dimension $n\in\mathbb N$. What is the Taylor expansion of the volume of a small ball around a point as a function of its radius?

The answer, at least up to the first nontrivial term, is:

\[\begin{align} \operatorname{vol}(B(x,r)) = \omega_n(r^n - \frac 1{6(n+2)}\operatorname{Sc}(x)r^{n+2}) + O(r^{n+4}), \end{align}\]

where $\operatorname{Sc}(x)$ is the scalar curvature at the point $x$ and $\omega_n$ is the volume of the unit ball of $\mathbb{R}^n$.

The proof

We prove this result by using normal coordinates around $x$. Let $v$ be any nonzero vector and define the path $\gamma(t) := tv$. Pick a vector $w$ and consider the vector field $J(t) := tw$ on $\gamma(t)$. It turns out that this vector field is a Jacobi vector field, so it satisfies the equation

\[\begin{align} J''(t) + R(\gamma(t), J(t))\gamma(t) = 0, \end{align}\]

where $R$ is the Riemannian curvature tensor. Using this equation, we may compute second-order Taylor expansion of the function

\[\begin{align} t \mapsto g_{\gamma(t)}(J(t), J(t)), \end{align}\]

and the result is

\[\begin{align} t^2 g_{tv}(w,w) = t^2\delta_{ij}w^iw^j - \frac{t^4}3 R_{iklj}v^kv^l w^iw^j + O(t^6). \end{align}\]

This implies

\[\begin{align} \label{eq:metricexpansion} g_v = \delta - \frac 13 R_{iklj} v^kv^l + O(v^4). \end{align}\]

For a matrix $A$, we have

\[\begin{align} \label{eq:matrixexpansion} \det(I - A) = 1 - \operatorname{tr}A + O(A^2), \end{align}\]

and taking determinants in \eqref{eq:metricexpansion} and then applying \eqref{eq:matrixexpansion} for $A_{ij} = \frac 13 R_{iklj}v^kv^l$ we obtain

\[\begin{align} \sqrt{\det g} = 1 - \frac 16 R_{kl}v^kv^l + O(v^4), \end{align}\]

where $R_{kl} = R^i_{kli}$ is the Ricci curvature tensor. It follows that for small $r$, we have

\[\begin{align} \operatorname{vol}(B_r(0)) = \omega_n(r^n - \frac 1{6(n+2)}\operatorname{Sc}(0)r^{n+2}) + O(r^{n+4}), \end{align}\]

where $\operatorname{Sc}(0)$ is the scalar curvature at $0$ and $\omega_n$ is the volume of the unit ball of $\mathbb{R}^n$ (see the next lemma. This result does not depend on the choice of coordinates unlike the previous expansion.

Lemma

For any square $n\times n$ matrix $A$ and $n$-dimensional ball of radius $r>0$ at the origin $B(r)$, we have

\[\begin{align} \int_{B(r)} \langle Ax, x\rangle dx & = \frac{1}{n+2}\operatorname{tr}(A) \cdot \operatorname{vol}(B(1))\cdot r^{n+2}. \end{align}\]

Proof

The off-diagonal terms cancel by the symmetry of the domain of integration,

\[\begin{align} \int_{B(r)} \langle Ax, x\rangle dx & = \sum_{i=1}^n\int_{B(r)} A_{ii} (x^i)^2 dx \\ & = r^{n+2}\sum_{i=1}^n A_{ii} \int_{B(1)} (x^i)^2 dx. \end{align}\]

Using Folland1 (or simply integrating this expression), we obtain

\[\begin{align} \int_{B(1)} (x^i)^2dx & = \frac{1}{n+2}\cdot 2\frac{\Gamma(3/2)\Gamma(1/2)^{n-1}}{\Gamma(\frac{n+2}2)} \\ & = \frac{1}{n+2}\cdot \operatorname{vol}(B(1)). \end{align}\]

Jacobi fields

A Jacobi field is the derivative in the parameter $\tau$ of a parametrized family of geodesics $\gamma_\tau(t)$. Using normal coordinates $U$, the vector field

\[\begin{align} (x,\xi) = (tv, tw) \end{align}\]

for $v\in U$ and $w\in\mathbb{R}^n$ is a Jacobi field for small $t$.

  1. Folland, Gerald B. How to integrate a polynomial over a sphere. Amer. Math. Monthly 108 (2001), no. 5, 446–448. ↩