Real hypersurfaces of a complex space form

Deshmukh, Sharief

doi:10.1007/s00605-010-0269-x

Cited by 9 publications

(6 citation statements)

References 19 publications

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“…. , 2n, in our notations (see [1,4,7,9,14,16,20,26,28,31,36,37,38] for Hopf hypersurfaces and their classification). If M is not Hopf, such formulas are not exact, in the sense that the reminder terms do not vanish in general; moreover (as shown in [22], where also a second (k = 2) horizontal Minkowski formula has been written) being Hopf is only a sufficient condition for these formulas to hold.…”

Section: Introduction Definitions and Statement Of The Resultsmentioning

confidence: 99%

Horizontal Newton operators and high-order Minkowski formula

Guidi

Martino

2020

Commun. Contemp. Math.

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In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

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Section: Introduction Definitions and Statement Of The Resultsmentioning

confidence: 99%

Horizontal Newton operators and high-order Minkowski formula

Guidi

Martino

2020

Commun. Contemp. Math.

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“…However, unlike the case of hypersurfaces in complex space forms which have been intensively studied (see, e.g., [1,8,21,22,31,33,34,38,44]) and despite the fact that Sasakian spaces are linked in a natural way to Kähler spaces, the geometry of hypersurfaces in Sasakian manifolds was not investigated with the same fervency, except some studies on hypersurfaces of Sasakian space forms (see [25]). In this setting, a remarkable result was obtained by Watanabe (cf.…”

Section: Introductionmentioning

confidence: 99%

Hypersurfaces of a Sasakian manifold - revisited

et al. 2021

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We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .

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“…The geometry and topology of a K-contact manifold is studied by several authors (cf. [5][6][7][8][9][10][11][12][13][14][15][16][17]). In [12] and [13], the author has used surgery on K-contact manifolds to classify 5-dimensional compact simply connected K-contact manifolds and also provided several examples of compact K-contact manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…Jacobi-type vector fields are also used in studying real hypersurfaces of a complex space form (cf. [6], [9]). It is known that a Jacobi-type vector field u on a Riemannian manifold (M, g) satisfies the differential equation…”

Section: Introductionmentioning

confidence: 99%