Abstract
Let (M n, g) and (N n+1, G) be Riemannian manifolds. Let TM n and TN n+1 be the associated tangent bundles. Let f: (M n, g) → (N n+1,G) be an isometrical immersion with \(g = f*G,F = (f,df):(TM^n ,\bar g) \to (TN^{n + 1} ,G_s )\) be the isometrical immersion with \(\bar g = F*G_s\) where (df) x : T x M → T f(x) N for any x ∈ M is the differential map, and G s be the Sasaki metric on TN induced from G. This paper deals with the geometry of TM n as a submanifold of TN n+1 by the moving frame method. The authors firstly study the extrinsic geometry of TM n in TN n+1. Then the integrability of the induced almost complex structure of TM is discussed.
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This work was supported by the National Natural Science Foundation of China (No. 61473059) and the Fundamental Research Funds for the Central University (No.DUT11LK47).
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Hou, Z., Sun, L. On the tangent bundle of a hypersurface in a Riemannian manifold. Chin. Ann. Math. Ser. B 36, 579–602 (2015). https://doi.org/10.1007/s11401-015-0936-2
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DOI: https://doi.org/10.1007/s11401-015-0936-2