Download Geometry of Cauchy-Riemann Submanifolds by Sorin Dragomir, Mohammad Hasan Shahid, Falleh R. Al-Solamy PDF
By Sorin Dragomir, Mohammad Hasan Shahid, Falleh R. Al-Solamy
This e-book gathers contributions through revered specialists at the conception of isometric immersions among Riemannian manifolds, and specializes in the geometry of CR constructions on submanifolds in Hermitian manifolds. CR buildings are a package deal theoretic recast of the tangential Cauchy–Riemann equations in complicated research regarding numerous complicated variables. The publication covers a variety of themes reminiscent of 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 came across the proposal of a CR submanifold of a Hermitian manifold in 1978, the ebook presents an up to date assessment of a number of issues within the geometry of CR submanifolds. offering unique info at the latest advances within the zone, it represents an invaluable source for mathematicians and physicists alike.
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Extra resources for Geometry of Cauchy-Riemann Submanifolds
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In this article, we denote a complete simply connected complex m-dimensional complex space form of constant holomorphic sectional curvature 4c ˜ m (4c). -Y. Chen ˜ m (4c) satisfies The curvature tensor R˜ of a complex space form M ˜ R(U, V, W ) = c{ V, W U − X, W V + J V, W JU − JU, W J V + 2 U, J V J W }. 7) ˜ m (4c) is holomorphically isometric to the complex projective It is well-known that M m m-space CP (4c), the complex Euclidean m-space Cm , or the complex hyperbolic m-space CH m (4c) according to c > 0, c = 0, or c < 0, respectively.
Sahin proved the following. 21 ([42]) There do not exist doubly warped product CR-submanifolds which are not (singly) warped product CR-submanifolds in the form f1 MT ×f2 M⊥ , where MT is a holomorphic submanifold and M⊥ is a totally real submanifold of a ˜ Kaehler manifold M. -Y. 7 CR-Warped Products with Compact Holomorphic Factor When the holomorphic factor NT of a CR-warped product NT ×f N⊥ is compact, we have the following sharp results. 22 ([24]) Let NT ×f N⊥ be a CR-warped product in the complex projective m-space CPm (4) of constant holomorphic sectional curvature 4.
And dim N⊥ > 1, then ||σ||2 = 2p ||∇ T ln λ||2 (3) If M is anti-holomorphic in M holds identically if and only if NT is a totally geodesic submanifold and N⊥ is a ˜ totally umbilical submanifold of M. For mixed foliate twisted product CR-submanifolds of Kaehler manifolds, we have the following result. 36 ([11]) Let M = NT ×λ N⊥ be a twisted product CR-submanifold of ˜ such that N⊥ is a totally real submanifold and NT is a holoa Kaehler manifold M ˜ If M is mixed totally geodesic, then we have morphic submanifold of M.