The Degree of Symmetry of Certain Compact Smooth Manifolds II*

Abstract

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361–380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of ℂP ² × V , where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.

Keywords

Degree of symmetry / Fiber bundle / Cobordism / Non-positive Curvature / Harmonic map / First cohomology / 57S15 / 53C44

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