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Jens Heber Gerhard Knieper Hemangi M. Shah 《Proceedings of the American Mathematical Society》2007,135(3):845-849
Let be a Hadamard manifold of dimension whose sectional curvature satisfies and whose curvature tensor satisfies for suitable constants and . We show that is of constant sectional curvature provided is asymptotically harmonic. This was previously only known if admits a compact quotient.
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For a non-compact, complete and simply connected manifoldM without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres inM is a radial function, then the geodesic spheres are convex. We also show that ifM is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on
it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index
lemma. 相似文献
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For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C
topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces. 相似文献
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