Abstract
Motivated by some issues which enter into the Gauss-Bonnet-Chern theorem in Finsler geometry, this paper studies the unit tangent sphere (or indicatrix) Ix M at each point x of a Pinsler manifold M. We demonstrate that the volume of ImM, calculated with respect to a Riemannian metric induced naturally by the Finsler structure, is in general a function of x. This contrasts sharply with the situation in Riemannian geometry. We also express the derivative of such volume function in terms of the second curvature tensor of the Chern connection. In particular, we find that this function is constant on Landsberg spaces (though that constant need not be equal to the value taken by Riemannian manifolds).
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Research partially supported by NSF Grant DMS 94-04097
Research partially supported by NSF Grant DMS 93-04731
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Bao, D., Shen, Z. On the Volume of Unit Tangent Spheres in a Pinsler Manifold. Results. Math. 26, 1–17 (1994). https://doi.org/10.1007/BF03322283
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DOI: https://doi.org/10.1007/BF03322283