On the Tangent Bundle of a Hypersurface in a Riemannian Manifold

Let(Mn, g) and(Nn+1, G) be Riemannian manifolds. Let TMn and TNn+1 be the associated tangent bundles. Let f :(Mn,g) →(Nn+1,G) be an isometrical immersion with g = f*G, F =(f, df) :(TMn, ■) →(TNn+1, Gs) be the isometrical immersion with ■= F*Gs where (df)x: TxM → Tf(x)N for any x ∈M is the differential map, and Gs be the Sasaki metric on TN induced from G. This paper deals with the geometry of TMn as a submanifold of TNn+1 by the moving frame method. The authors firstly study the extrinsic geometry of TMn in TNn+1. Then the integrability of the induced almost complex structure of TM is discussed.

Let （Mn, g） and （N^n＋1, G） be Riemannian manifolds. Let TMn and TN^n＋1 be the associated tangent bundles. Let f ： （M^n, g） → （N^＋1, G） be an isometrical immersion with g = f^＊G, F = （f, df） ： （TM^n,g） → （TN^n＋1, Gs） be the isometrical immersion with g= F＊Gs where （df）x ： TxM → Tf（x）N for any x∈ M is the differential map, and Gs 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 TMn in TN^n＋1. Then the integrability of the induced almost complex structure of TM is discussed.