Einstein manifolds of non-negative sectional curvature and entropy

We show that if $(M^{n},g)$ is a closed Einstein manifold of non-negative curvature then $-\log\,R\leq \frac{\pi\,\sqrt{n-1}\,(n-2)}{2},$ where $R$ is the radius of convergence of the series $\sum_{i\geq 2}\dim \,(\pi_{i}(M)\otimes$\,{\test Q}\hspace{-.1mm})\hspace{.4mm}$t^{i}.$ If we suppose in addition that $M$ is formal then we show that: $$\dim\, H_{*}(M,{\text{\test Q}})\leq \left[1+\exp\left(\frac{\pi\,\sqrt{n-1}\,(n-2)}{2}\right)\right]^{n}.$$ These results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow of $g$ in terms of the curvature tensor.

Full Text (PDF format)

Published 1 January 2000