On submersions of the complex indicatrix

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5081-5092
Author(s):  
Elena Popovicia
Keyword(s):  
Fixed Point ◽  
Complex Projective Space ◽  
Finsler Space ◽  
Hermitian Manifold ◽  
Almost Hermitian Manifold ◽  
Codimension One ◽  
Real Submanifold ◽  
Fundamental Equations ◽  
Complex Projective ◽  
Extrinsic Sphere

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

scholarly journals Regular cyclic actions on complex projective space with codimension-two fixed points

1998 ◽  
Vol 65 (1) ◽  
pp. 51-67 ◽  
Author(s):  
Robert D. Little
Keyword(s):  
Fixed Point ◽  
Projective Space ◽  
Fixed Points ◽  
Cyclic Group ◽  
Complex Projective Space ◽  
Fixed Point Set ◽  
Codimension Two ◽  
Point Set ◽  
Isolated Point ◽  
Complex Projective

AbstractIfM2nis a cohomologyCPnandPis an odd prime, letGpbe the cyclic group of orderp. A TypeI I0Gpaction onM2nis an action with fixed point set a codimension-2 submanifold and an isolated point. A TypeI I0Gpaction is standard if it is regular and the degree of the fixed codimension-2 submanifold is one. If n is odd and M2nadmits a standardGpaction of TypeI I0, then every TypeI I0GpactionM2nis standard and so, if n is odd,CPnadmits aGpaction of TypeI I0if and only if the action is standard.

scholarly journals Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

2015 ◽  
Vol 26 (05) ◽  
pp. 1550036 ◽  
Author(s):  
Jiří Lebl
Keyword(s):  
Algebraic Curve ◽  
Complex Projective Space ◽  
First Integral ◽  
Holomorphic Foliation ◽  
Singular Set ◽  
Analytic Subvariety ◽  
Codimension One ◽  
Real Analytic ◽  
Complex Cone ◽  
Complex Projective

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

scholarly journals From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space

2002 ◽  
Vol 66 (3) ◽  
pp. 465-475 ◽  
Author(s):  
J. Bolton ◽  
C. Scharlach ◽  
L. Vrancken
Keyword(s):  
Projective Space ◽  
Minimal Surface ◽  
Complex Projective Space ◽  
Lagrangian Submanifold ◽  
3 Dimensional ◽  
Ellipse Of Curvature ◽  
Complex Projective

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

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