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 denote a complete simply connected manifold of nonpositive sectional curvature. For each point let denote the diffeomorphism of that fixes and reverses all geodesics through . The symmetry diffeomorphism group generated by all diffeomorphisms extends naturally to group of homeomorphisms of the boundary sphere . A subset of is called *involutive* if it is invariant under .

Theorem. Let *be a proper, closed involutive subset. For each point* *let* *denote the linear span in* *of those vectors at* *that are tangent to a geodesic* *whose asymptotic equivalence class* *belongs to* . *If* *is a proper subspace of* *for some point* , *then* *splits as a Riemannian product* *such that* *is the distribution of* *induced by* .

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 is irreducible and admits a proper, closed involutive subset , then is isometric to a symmetric space of noncompact type and rank .

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0957076-1

Article copyright:
© Copyright 1988
American Mathematical Society