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Transactions of the American Mathematical Society

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Symmetry diffeomorphism group of a manifold of nonpositive curvature


Author: Patrick Eberlein
Journal: Trans. Amer. Math. Soc. 309 (1988), 355-374
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9947-1988-0957076-1
MathSciNet review: 957076
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Abstract: Let $ \tilde M$ denote a complete simply connected manifold of nonpositive sectional curvature. For each point $ p \in \tilde M$ let $ {s_p}$ denote the diffeomorphism of $ \tilde M$ that fixes $ p$ and reverses all geodesics through $ p$. The symmetry diffeomorphism group $ {G^{\ast}}$ generated by all diffeomorphisms $ \{ {s_p}:\,p \in \tilde M\} $ extends naturally to group of homeomorphisms of the boundary sphere $ \tilde M(\infty )$. A subset $ X$ of $ \tilde M(\infty )$ is called involutive if it is invariant under $ {G^{\ast}}$.

Theorem. Let $ X \subseteq \tilde M(\infty )$ be a proper, closed involutive subset. For each point $ p \in \tilde M$ let $ N(p)$ denote the linear span in $ {T_p}\tilde M$ of those vectors at $ p$ that are tangent to a geodesic $ \gamma $ whose asymptotic equivalence class $ \gamma (\infty )$ belongs to $ X$. If $ N(p)$ is a proper subspace of $ {T_p}\tilde M$ for some point $ p \in \tilde M$, then $ \tilde M$ splits as a Riemannian product $ {\tilde M_1} \times {\tilde M_2}$ such that $ N$ is the distribution of $ \tilde M$ induced by $ {\tilde M_1}$.

This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if $ \tilde M$ is irreducible and $ \tilde M(\infty )$ admits a proper, closed involutive subset $ X$, then $ \tilde M$ is isometric to a symmetric space of noncompact type and rank $ k \geqslant 2$.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0957076-1
Article copyright: © Copyright 1988 American Mathematical Society

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