Holomorphic Distribution Research Articles

A characterization of pseudo-Einstein real hypersurfaces of a complex space form

We give a characterization of pseudo-Einstein real hypersurfaces of a complex space form. We prove that if the Ricci tensor S of a real hypersurface M of a complex space form with constant principal curvatures satisfies for any , where a is a function and T 0 is the holomorphic distribution on M , then M is a pseudo-Einstein real hypersurface.

Integrability of distributions in GCR-lightlike submanifolds of indefinite cosymplectic manifolds

We give necessary and sufficient conditions for the integrability of various distributions of GCR-lightlike submanifold of an indefinite Cosymplectic manifold.We also find the conditions for each leaf of holomorphic distribution and radical distribution to be totally geodesic in submanifold.

The complex Goldberg–Sachs theorem in higher dimensions

We study the geometric properties of holomorphic distributions of totally null m-planes on a (2m+ϵ)-dimensional complex Riemannian manifold (M,g), where ϵ∈{0,1} and m≥2. In particular, given such a distribution N, say, we obtain algebraic conditions on the Weyl tensor and the Cotton–York tensor which guarantee the integrability of N, and in odd dimensions, of its orthogonal complement. These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg–Sachs theorem from four to higher dimensions.Higher-dimensional analogues of the Petrov type D condition are defined, and we show that these lead to the integrability of up to 2m holomorphic distributions of totally null m-planes. Finally, we adapt these findings to the category of real smooth pseudo-Riemannian manifolds, commenting notably on the applications to Hermitian geometry and Robinson (or optical) geometry.

Real hypersurfaces in a complex projective space with pseudo- % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv % 39gaiyaacqWFdcpraaa!47B8! $$ \mathbb{D} $$ -parallel structure Jacobi operator

We introduce the new notion of pseudo-\( \mathbb{D} \)-parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.

A characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M n(c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T 0(x), a being a function, where T 0 is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.

Residues of codimension one singular holomorphic distributions

For a codimension one locally-free singular holomorphic distribution, we give a residue formula in terms of the conormal sheaf given by Pfaffian equations. We also prove a Baum-Bott type residue formula for singular distributions.

Ricci Recurrent $CR$ Submanifolds of a Complex Space Form

We show that there is no $CR$ submanifold with semi-flat normal connection and with recurrent Ricci tensor in a complex space form of nonzero constant holomorphic sectional curvature, if the dimension of its holomorphic distribution is greater than 2.

Characterizations of Real Hypersurfaces in a Complex Space Form

AbstractWe study a real hypersurface M in a complex space form Mn(c), c ≠ 0, whose shape operator and structure tensor commute each other on the holomorphic distribution of M.

CHARACTERIZATIONS OF REAL HYPERSURFACES OF COMPLEX SPACE FORMS IN TERMS OF RICCI OPERATORS

We prove that a real hypersurface M in a complex space form Mn(c), <TEX>$c{\neq}0$</TEX>, whose Ricci operator and structure tensor commute each other on the holomorphic distribution and the Ricci operator is <TEX>${\eta}-parallel$</TEX>, is a Hopf hypersurface. We also give a characterization of this hypersurface.

Real hypersurfaces in complex projective space whose structure Jacobi operator is $\mathbb D$-parallel

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is parallel in any direction of the maximal holomorphic distribution.

ON CHARACTERIZATIONS OF REAL HYPERSURFACES WITH ${\eta}-PARALLEL$ RICCI OPERATORS IN A COMPLEX SPACE FORM

We shall give a characterization of a real hypersurface M in a complex space form Mn(c), , whose Ricci operator and structure tensor commute each other on the holomorphic distribution of M, and the Ricci operator is .

Holomorphicity and the Walczak formula on Sasakian manifolds

The Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. M. Svensson [Holomorphic foliations, harmonic morphisms and the Walczak formula, J. London Math. Soc. (2) 68 (3) (2003) 781–794] has shown that this formula simplifies to a Bochner-type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. We show in this paper that such results have a counterpart in Sasakian geometry. To this end, we build on a theory of (contact) holomorphicity on almost contact metric manifolds. Some other applications for (pseudo-)harmonic morphisms on Sasaki manifolds are outlined.

HOLOMORPHIC FOLIATIONS, HARMONIC MORPHISMS AND THE WALCZAK FORMULA

A formula of Walczak is applied to two situations in differential geometry. Holomorphic distributions on Kahler manifolds are studied, and it is shown how the formula simplifies to a Bochner type formula, which is particularly useful in the study of integrable distributions. Then the Walczak formula is applied in the context of harmonic morphisms, where it provides a means of investigating the vertical Laplacian of the dilation. It is shown that, under some additional conditions on the map and the domain, the p-energy is infinite for p sufficiently large.

On holomorphic one-forms transverse to closed hypersurfaces

In this note we announce some achievements in the study of holomorphic distributions admitting transverse closed real hypersurfaces. We consider a domain with smooth boundary in the complex affine space of dimension two or greater. Assume that the domain satisfies some cohomology triviality hypothesis (for instance, if the domain is a ball). We prove that if a holomorphic one form in a neighborhood of the domain is such that the corresponding holomorphic distribution is transverse to the boundary of the domain then the Euler-Poincaré-Hopf characteristic of the domain is equal to the sum of indexes of the one-form at its singular points inside the domain. This result has several consequences and applies, for instance, to the study of codimension one holomorphic foliations transverse to spheres.

Deformation rigidity of the rational homogeneous space associated to a long simple root

As a continuation of our previous works we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces G/ P of Picard number 1, confirming its validity whenever G/ P is associated to a long simple root. For these rational homogeneous spaces the minimal G-invariant holomorphic distribution D is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of G/ P is completely determined by the nilpotent symbol algebra of the weak derived differential system of D. The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point o∈ G/ P we prove that the nilpotent symbol algebra at o is nothing other than the universal Lie algebra generated by D o subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra G.

Geometry of Holomorphic Distributions of Real Hypersurfaces in a Complex Projective Space

We characterize homogeneous real hypersurfaces M's of type (A1), (A2) and (B) of a complex projective space in the class of real hypersurfaces by studying the holomorphic distribution T0M of M.

Sasakian anti-holomorphic submanifolds of a Kaehler manifold

Let $ M$ be a connected Sasakian anti-holomorphic submanifold of a Kaehler mnifold with flat norml connection and with dim $ D\ge 4$, where $ D$ is the holomorphic distribution on $ M$. We show that $ M$ is locally Riemannian product $ M' \times M''$ where $ M'$ is homothetic to a Sasakian manifold and $ M''$ is a locally Euclidean space.

On some integrable generalizations of the continuous Toda system

In the present paper, we obtain some integrable generalizations of the continuous Toda system generated by a flat connection form taking values in higher grading subspaces of the algebra of the area-preserving diffeomorphism of the torus T2 and construct their general solutions. The grading condition we use here, imposed on the connection, can be realized in terms of holomorphic distributions on the corresponding homogeneous spaces.

CR-SUBMANIFOLDS OF A QUASl-KAEHLER MANIFOLD

&#x0D; &#x0D; &#x0D; Let $M$ be a CR-submanifold of a quasi-Kaehler manifold $N$. Sufficient conditions for the holomorphic distribution $D$ in $M$ to be integrable are derived. We also show that $D$ is minimal. It follows that an (almost) complex submanifold of a quasi-Kaehler manifold is minimal, this generalizes the well known result that a complex submanifold of a Kaehler manifold is minimal. &#x0D; &#x0D; &#x0D;

On CR‐submanifolds of the six‐dimensional sphere

We consider proper CR‐submanifolds of the six‐dimensional sphere S6. We prove that S6 does not admit compact proper CR‐submanifolds with non‐negative sectional curvature and integrable holomorphic distribution.