Illinois Journal of Mathematics

On the geometry of positively curved manifolds with large radius

Qiaoling Wang

Full-text: Open access

Abstract

Let $M$ be an $n$-dimensional complete connected Riemannian manifold with sectional curvature $K_M\geq 1$ and radius $\operatorname{rad}(M)>\pi /2$. For any $x\in M$, denote by $\operatorname{rad} (x)$ and $\rho (x)$ the radius and conjugate radius of $M$ at $x$, respectively. In this paper we show that if $\operatorname{rad} (x)\leq \rho (x)$ for all $x\in M$, then $M$ is isometric to a Euclidean $n$-sphere. We also show that the radius of any connected nontrivial (i.e., not reduced to a point) closed totally geodesic submanifold of $M$ is greater than or equal to that of $M$.

Article information

Source
Illinois J. Math., Volume 48, Number 1 (2004), 89-96.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258136175

Digital Object Identifier
doi:10.1215/ijm/1258136175

Mathematical Reviews number (MathSciNet)
MR2048216

Zentralblatt MATH identifier
1048.53024

Subjects
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Citation

Wang, Qiaoling. On the geometry of positively curved manifolds with large radius. Illinois J. Math. 48 (2004), no. 1, 89--96. doi:10.1215/ijm/1258136175. https://projecteuclid.org/euclid.ijm/1258136175


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