By Sorin Dragomir, Mohammad Hasan Shahid, Falleh R. Al-Solamy
This e-book gathers contributions through revered specialists at the thought of isometric immersions among Riemannian manifolds, and makes a speciality of the geometry of CR buildings on submanifolds in Hermitian manifolds. CR constructions are a package deal theoretic recast of the tangential Cauchy–Riemann equations in advanced research related to a number of complicated variables. The e-book covers a variety of issues akin to Sasakian geometry, Kaehler and in the neighborhood 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 came across the proposal of a CR submanifold of a Hermitian manifold in 1978, the ebook offers an up to date review of numerous themes within the geometry of CR submanifolds. proposing special info at the latest advances within the quarter, it represents an invaluable source for mathematicians and physicists alike.
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This ebook gathers contributions through revered specialists at the concept 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 regarding a number of complicated variables.
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Extra resources for Geometry of Cauchy-Riemann Submanifolds
Zm ) = (λz0 , . . , λzm ) for λ ∈ C∗ . Let π(z) denote the equivalent class containing z under this action. Then the set of equivalent classes is the complex projective m-space CPm (4) with the complex structure induced from the complex structure on C∗m+1 . For any two natural numbers h and p, we define a map: φ˘ : C∗h+1 × S p (1) → C∗h+p+1 by ˘ 0 , . . , zh ; w0 , . . , w p ) = w 0 z0 , w1 z0 , . . , w p z0 , z1 , . . , zh φ(z for (z0 , . . , zh ) in C∗h+1 and (w0 , . . , wp ) in S p with p j=0 wj2 = 1.
Growth estimates for warping functions and their geometric applications. Glasg. Math. J. 51(3), 579–592 (2009) 36. : Nearly Kähler manifolds. J. Differ. Geom. 4, 283–309 (1970) 37. : Generic warped product submanifolds in a Kaehler manifold. Filomat 22(1), 139–144 (2008) 38. : Isometric immersions of Riemannian products. J. Differ. Geom. 5(1–2), 159–168 (1971) 39. : The imbedding problem for Riemannian manifolds. Ann. Math. 63, 20–63 (1956) 40. : Semi-Riemannian Geometry with Applications to Relativity.
Zh ; w0 , . . , w p ) = w 0 z0 , w1 z0 , . . , w p z0 , z1 , . . , zh φ(z for (z0 , . . , zh ) in C∗h+1 and (w0 , . . , wp ) in S p with p j=0 wj2 = 1. -Y. Chen Since the image of φ˘ is invariant under the action of C∗ , the composition: ˘ φ π π ◦ φ˘ : C∗h+1 × S p − → C∗h+p+1 − → CPh+p (4) induces a CR-immersion of the product manifold NT × S p into CPh+p (4), where NT = (z0 , . . , zh ) ∈ CPh : z0 = 0 is a proper open subset of CPh (4). Clearly, the induced metric on NT × S p is a warped product metric and the holomorphic factor NT is non-compact.