By Sorin Dragomir, Mohammad Hasan Shahid, Falleh R. Al-Solamy

This booklet gathers contributions by means of revered specialists at the idea of isometric immersions among Riemannian manifolds, and specializes in the geometry of CR constructions on submanifolds in Hermitian manifolds. CR constructions are a package deal theoretic recast of the tangential Cauchy–Riemann equations in advanced research concerning a number of advanced variables. The publication covers a variety of themes akin to Sasakian geometry, Kaehler and in the community conformal Kaehler geometry, the tangential CR equations, Lorentzian geometry, holomorphic statistical manifolds, and paraquaternionic CR submanifolds.

Intended as a tribute to Professor Aurel Bejancu, who chanced on the inspiration of a CR submanifold of a Hermitian manifold in 1978, the publication presents an up to date review of a number of subject matters within the geometry of CR submanifolds. featuring designated info at the latest advances within the sector, it represents an invaluable source for mathematicians and physicists alike.

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**Extra resources for Geometry of Cauchy-Riemann Submanifolds**

**Example 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.