Let $M$ be an $n$-dimensional complete Riemannian manifold with Riccicurvature $\ge n-1$. In \cite{colding1, colding2}, Tobias Colding, bydeveloping some new techniques, proved that the following three condtions: 1)$d_{GH}(M, S^n)\to 0$; 2) the volume of $M$${\text{Vol}}(M)\to{\text{Vol}}(S^n)$; 3) the radius of $M$${\text{rad}}(M)\to\pi$ are equivalent. In \cite{peter}, Peter Petersen, bydeveloping a different technique, gave the 4-th equivalent condition, namely heproved that the $n+1$-th eigenvalue of $M$ $\lambda_{n+1}(M)\to n$ is alsoequivalent to the radius of $M$ ${\text{rad}}(M)\to\pi$, and hence the othertwo. In this note, we give a new proof of Petersen's theorem by utilizingColding's techniques.
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机译：令$ M $是具有Riccicurvature $ \ ge n-1 $的$ n $维完整的黎曼流形。在\ cite {colding1，colding2}中，Tobias Colding通过开发一些新技术，证明了以下三个条件：1）$ d_ {GH}（M，S ^ n）\至0 $； 2）$ M $$ {\ text {Vol}}（M）\ to {\ text {Vol}}（S ^ n）$的数量； 3）$ M $$ {\ text {rad}}（M）\ to \ pi $的半径相等。在\ cite {peter}中，彼得·彼得森（Peter Petersen）通过开发另一种技术，给出了第4个等价条件，即证明了$ M $ $ \ lambda_ {n + 1}（M）的第n + 1 $个特征值\ to n $也等于$ M $ $ {\ text {rad}}（M）\ to \ pi $的半径，因此也等于。在本文中，我们通过利用Colding的技术为Petersen定理提供了新的证明。
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