In this paper, we estimate the degree of symmetry and the semi-simple degree of symmetry of certain fiber bundles by virtue of the rigidity theorem with respect to the harmonic map due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of certain product manifolds. In addition, by Albanese map, we estimate the degree of symmetry and the semi-simple degree of symmetry of a compact smooth manifold under some topological assumptions.

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