On the tangent bundle of a hypersurface in a Riemannian manifold

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.

Keywords

Hypersurfaces / Tangent bundle / Mean curvature vector / Sasaki metric / Almost complex structure / Kählerian form

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