By Shoshichi Kobayashi

This two-volume advent to differential geometry, a part of Wiley's renowned Classics Library, lays the basis for knowing a space of research that has turn into important to modern arithmetic. it truly is thoroughly self-contained and may function a reference in addition to a educating advisor. quantity 1 provides a scientific creation to the sphere from a short survey of differentiable manifolds, Lie teams and fibre bundles to the extension of neighborhood adjustments and Riemannian connections. the second one quantity maintains with the research of variational difficulties on geodesics via differential geometric facets of attribute sessions. either volumes familiarize readers with easy computational ideas.

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**Extra info for Foundations of Differential Geometry (Wiley Classics Library) (Volume 2)**

**Sample text**

A curve γ = γ(t) in a contact manifold is called a Legendre curve if η(β (t)) = 0 along β. Let S 2n+1 (c) denote the hypersphere in Cn+1 with curvature c centered at the origin. Then S 2n+1 (c) is a contact manifold endowed with a canonical contact structure which is the dual 1-form of the characteristic vector field Jξ, where J is the complex structure and ξ the unit normal vector on S 2n+1 (c). g. a diffeomorphism of a contact manifold is a contact transformation if and only if it maps Legendre curves to Legendre curves.

23) holds identically if and only if the following four statements hold: ˜ (a) NT is a totally geodesic holomorphic submanifold of M; ˜ with −∇(ln fi ) (b) For each i ∈ {2, . . , k}, Ni is a totally umbilical submanifold of M as its mean curvature vector; ˜ and (c) f2 N2 × · · · ×fk Nk is immersed as mixed totally geodesic submanifold in M; (d) For each point p ∈ N, the first normal space Im hp is a subspace of J(Tp N⊥ ). Remark 1 B. 13 to the following. 19 There exist no warped product submanifolds of the type Mθ ×f MT and MT ×f Mθ in a Kaehler manifold, where Mθ is a proper slant submanifold and ˜ MT is a holomorphic submanifold of M.

6 ([2]) Let N be an anti-holomorphic submanifold in a complex space ˜ 1+p (4c) with h = rank C D = 1 and p = rank D⊥ ≥ 2. Then, we have form M δ(D) ≤ (p − 1)(p + 2)2 2 p H + (p + 3)c. -Y. 46) holds identically if and only if c = 0 and either (i) N is a totally geodesic anti-holomorphic submanifold of Ch+p or, (ii) up to dilations and rigid motions, N is given by an open portion of the following product immersion: φ : C × S p (1) → C1+p ; (z, x) → (z, w(x)), z ∈ C, x ∈ S p (1), where w : S p (1) → Cp is the Whitney p-sphere.