Abstract
By using moving frame theory, first we introduce 2p-th mean curvatures and (2p + 1)-th mean curvature vector fields for a submanifold. We then give an integral expression of them that characterizes them as mean values of symmetric functions of principle curvatures. Next we apply it to derive directly the celebrated Weyl-Gray tube formula in terms of integrals of the 2p-th mean curvatures and some Minkowski-type integral formulas.
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This work was partially supported by the National Natural Science Foundation of China (Grant No. 107010007)
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Ge, J. On mean curvatures in submanifolds geometry. Sci. China Ser. A-Math. 51, 1127–1134 (2008). https://doi.org/10.1007/s11425-007-0182-5
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DOI: https://doi.org/10.1007/s11425-007-0182-5