\[ \DeclareMathOperator{\Ric}{Ric} \DeclareMathOperator{\Vol}{Vol} \]

Consider a connected Riemannian manifold \(M\) of dimension \(n\). Assume \(\Ric(M)>(n-1)k\) for some \(k\in \mathbb{R}\). We denote the Riemannian manifold with constant sectional curvature \(k\) as \(M_k\). Then for every \(p\in M\) and \(p_k\in M_k\) the function

\begin{equation*} \phi(r) = \frac{\Vol(B_r(p))}{\Vol(B_r(p_k))} \end{equation*}

is non-increasing on \((0,\infty)\).

Furthermore, we have

\begin{equation*} \phi(r)\xrightarrow{r \to 0^+} 1. \end{equation*}

[1]

References Link to heading

1. Bishop–Gromov inequality, Dec. 8, 2021. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Bishop%E2%80%93Gromov_inequality&oldid=1059331416 [Accessed: Dec. 18, 2024].