Conformal Foliations Research Articles

Harmonic morphisms on Lie groups and minimal conformal foliations of codimension two

Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This means that locally the leaves of F are fibres of a complex-valued harmonic morphism. In the Riemannian case, we prove that if the metric restricted to K is biinvariant then F is totally geodesic.

Natural almost Hermitian structures on conformally foliated 4-dimensional Lie groups with minimal leaves

Let (G, g) be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation mathcal {F} with minimal leaves. Let J be an almost Hermitian structure on G adapted to the foliation mathcal {F}. We classify such structures J which are almost Kähler (mathcal {A}mathcal {K}), integrable (mathcal {I}) or Kähler (mathcal {K}). Hereby we construct several new multi-dimensional examples in each class.

Biconformal equivalence between 3-dimensional Ricci solitons

Biconformal deformations in the presence of a conformal foliation by curves are exploited to study equivalence between 3-dimensional Ricci solitons. We show that a wide class of solitons are biconformally equivalent to the flat metric.

Four-dimensional Einstein metrics from biconformal deformations

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. One particular family of examples have ends that collapse asymptotically to ${\mathbb R}^2$.

Conformal foliations on Lie groups and complex-valued harmonic morphisms

We study left-invariant foliations F on Riemannian Lie groups G generated by a subgroup K. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations F when the subgroup K is one of the important SU(2)×SU(2), SU(2)×SL2(R), SU(2)×SO(2) or SL2(R)×SO(2). By this we yield new multi-dimensional families of Lie groups G carrying such foliations in each case. These foliations F produce local complex-valued harmonic morphisms on the corresponding Lie group G.

Dynamics of conformal foliations

Abstract The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

Critical Point Equation on Almost Kenmotsu Manifolds

We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the $CPE$ conjecture, then the manifold is Einstein.

Riemannian foliations with Ehresmann connection

It is shown that the structural theory of Molino for Riemannian foliations on compact manifolds and complete Riemannian manifolds may be generalized to a Riemannian foliations with Ehresmann connection. Within this generalization there are no restrictions on the codimension of the foliation and on the dimension of the foliated manifold. For a Riemannian foliation (M,F) with Ehresmann connection it is proved that the closure of any leaf forms a minimal set, the family of all such closures forms a singular Riemannian foliation (M,F¯¯¯¯). It is shown that in M there exists a connected open dense F¯¯¯¯-saturated subset M0 such that the induced foliation (M0,F¯¯¯¯|M0) is formed by fibers of a locally trivial bundle over some smooth Hausdorff manifold. The equivalence of some properties of Riemannian foliations (M,F) with Ehresmann connection is proved. In particular, it is shown that the structural Lie algebra of (M,F) is equal to zero if and only if the leaf space of (M,F) is naturally endowed with a smooth orbifold structure. Constructed examples show that for foliations with transversally linear connection and for conformal foliations the similar statements are not true in general.

The Fischer-Marsden conjecture on almost Kenmotsu manifolds

The purpose of this paper is to investigate the Fischer-Marsden conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional non-Kenmotsu (k, µ)′-almost Kenmotsu manifold satisfies the Fischer-Marsden conjecture, then the manifold is locally isometric to the product space ℍ2(−4) × ℝ. Further, we prove that if the metric of a complete almost Kenmotsu manifold with conformal Reeb foliation satisfies the Fischer-Marsden conjecture, then the manifold is Einstein provided the scalar curvature r ≠ −2n(2n + 1).

Physical measures for the geodesic flow tangent to a transversally conformal foliation

We consider a transversally conformal foliation F of a closed manifold M endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either there is a transverse holonomy-invariant measure for F, or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin covers a set full for the Lebesgue measure. We also give necessary and sufficient conditions for the foliated geodesic flow to be partially hyperbolic in the case where the foliation is transverse to a projective circle bundle over a closed hyperbolic surface.

Certain results on almost Kenmotsu manifolds with conformal reeb foliation

Certain results on almost Kenmotsu manifolds with conformal reeb foliation

Harmonic morphisms from 5-dimensional Lie groups

We consider 5-dimensional Lie groups equipped with a left-invariant Riemannian metric. On such groups we construct left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce complex-valued harmonic morphisms locally defined on the Lie group.

Transverse Equivalence of Complete Conformal Foliations

We study the problem of classification of complete non-Riemannian conformal foliations of codimension q > 2 with respect to transverse equivalence. It is proved that two such foliations are transversally equivalent if and only if their global holonomy groups are conjugate in the group of conformal transformations of the q-dimensional sphere Conf ( $$ {\mathbb{S}}^{\mathrm{q}} $$ ). Moreover, any countable essential subgroup of the group Conf ( $$ {\mathbb{S}}^{\mathrm{q}} $$ ) is realized as the global holonomy group of some non-Riemannian conformal foliation of codimension q. Bibliography: 16 titles.

Holomorphic harmonic morphisms from cosymplectic almost Hermitian manifolds

We study 4-dimensional orientable Riemannian manifolds equipped with a minimal and conformal foliation $${\mathcal {F}}$$ of codimension 2. We prove that the two adapted almost Hermitian structures $$J_1$$ and $$J_2$$ are both cosymplectic if and only if $${\mathcal {F}}$$ is Riemannian and its horizontal distribution $${\mathcal {H}}$$ is integrable.

Holomorphic harmonic morphisms from four-dimensional non-Einstein manifolds

We construct four-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not Kähler. We then prove that the Riemannian Lie groups constructed are not Einstein manifolds. This answers an important open question in the theory of complex-valued harmonic morphisms from Riemannian 4-manifolds.

Harmonic morphisms from four-dimensional Lie groups

We consider 4-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.

On existence of regular jacobi structures

We prove \(h\)-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We then interpret the results in terms of regular Jacobi structures on manifolds.

CR geometry and conformal foliations

We use the CR geometry of the standard hyperquadric in $$\mathbb{CP }_3$$ to give a detailed twistor description of conformal foliations in Euclidean 3-space.

Global attractors of complete conformal foliations

We prove that every complete conformal foliation of codimension is either Riemannian or a -foliation. We further prove that if is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In this theorem we do not assume that the manifold is compact. In particular, every proper conformal non-Riemannian foliation has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all nonclosed leaves is a connected -dimensional orbifold. We show that every countable group of conformal transformations of the sphere can be realized as the global holonomy group of a complete conformal foliation. Examples of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed.Bibliography: 20 titles.

Ray congruences that generate conformal foliations

Given a Riemannian 3-manifold (M3, g0) endowed with a unit vector field U0 that is tangent to a conformal foliation, we require that the pair extend to a space-time \documentclass[12pt]{minimal}\begin{document}$({\mathcal M}, {\mathcal G})$\end{document}(M,G) endowed with a space-like unit vector field Ut in such a way that Ut simultaneously generates null geodesics and is tangent to a conformal foliation on space-like slices t = constant. The property that the foliation be conformal is intimately related to CR-geometry.