By Sorin Dragomir, Mohammad Hasan Shahid, Falleh R. Al-Solamy
This e-book gathers contributions via revered specialists at the thought 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 advanced research regarding a number of advanced variables. The e-book covers quite a lot of themes resembling 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 found the idea of a CR submanifold of a Hermitian manifold in 1978, the publication presents an updated evaluate of numerous subject matters within the geometry of CR submanifolds. providing specific details at the most up-to-date advances within the quarter, it represents an invaluable source for mathematicians and physicists alike.
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Additional resources for Geometry of Cauchy-Riemann Submanifolds
24) holds identically if and only if we have: (1) The warping function f is constant. (2) (NT , gNT ) is holomorphically isometric to CPh (4) and it is isometrically immersed in CPm as a totally geodesic complex submanifold. (3) (N⊥ , f 2 gN⊥ ) is isometric to an open portion of the real projective p-space RPp (1) of constant sectional curvature one and it is isometrically immersed in CPm as a totally geodesic totally real submanifold. (4) NT ×f N⊥ is immersed linearly fully in a complex subspace CPh+p+hp (4) of CPm (4); and moreover, the immersion is rigid.
32 () Suppose that a is a positive number and γ(t) = ( 1 (t), 2 (t)) is a unit speed Legendre curve γ : I → S 3 (a2 ) ⊂ C2 defined on an open interval I. Let x : S∗2n+1 × I → Cn+2 be the map defined by n x(z0 , . . , zn , t) = a 1 (t)z0 , a 2 (t)z0 , z1 , . . , zn , zk z¯k = 1. 33) k=0 Then (1) x induces an isometric immersion ψ : S∗2n+1 (1) ×a|z0 | I → S 2n+3 (1). (2) The image ψ(S∗2n+1 (1) ×a|z0 | I) in S 2n+3 (1) is invariant under the action of U(1). (3) the projection ψπ : π(S∗2n+1 (1) ×a|z0 | I) → CPn+1 (4) of ψ via π is a warped product hypersurface CP0n ×a|z0 | I in CPn+1 (4).
16(2), 105–121 (2000) 12. : Riemannian submanifolds. , Verstraelen, L. ) Handbook of Differential Geometry, vol. I, pp. 187–418. North Holland Publishing, Amsterdam (2000) 13. : Complex extensors, warped products and Lagrangian immersions. Soochow J. Math. 26(1), 1–18 (2000) 14. : Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133(3), 177–195 (2001) 15. : Geometry of warped product CR-submanifolds in Kaehler manifolds. II. Monatsh. Math. 134(2), 103–119 (2001) 16.