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We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere S 3 . We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on S 3 , whose integral curves are great circles, is a Hopf vector field. [ABSTRACT FROM AUTHOR]

We expand upon the notion of a pre-section for a singular Riemannian foliation (M , F) , i.e. a proper submanifold N ⊂ M retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of (M , F) and the leaf space M / F . In particular, we show that if a foliated manifold M has positive sectional curvature and contains a non-trivial pre-section, then the leaf space M / F has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions. [ABSTRACT FROM AUTHOR]

In this paper, we study the twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem on a transverse Kähler foliation. And we give some properties for Δ κ -harmonic forms and prove the Kodaira–Serre-type duality and Dolbeault isomorphism for the twisted basic Dolbeault cohomology. [ABSTRACT FROM AUTHOR]

Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors in Baudoin and Grong (Ann Glob Anal Geom 56(2):403–428, 2019). [ABSTRACT FROM AUTHOR]

We describe a local model for any singular Riemannian foliation in a neighborhood of a closed saturated submanifold of a singular stratum. Moreover, we construct a Lie groupoid that controls the transverse geometry of the linear approximation of the singular Riemannian foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid. [ABSTRACT FROM AUTHOR]

In this paper, we discuss two Liouville-type theorems for subelliptic harmonic maps from sub-Riemannian manifolds to Riemannian manifolds. One is the Dirichlet version which states that two subelliptic harmonic maps from a sub-Riemannian manifold with boundary to a regular ball must be same if their restrictions on boundary are same; it is generalized to complete noncompact domains as well. The other is the vanishing-type theorem for finite L p -energy subelliptic harmonic maps on complete noncompact totally geodesic Riemannian foliations which are special sub-Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a new structure on a Riemannian manifold: a distribution represented as the sum of k > 2 pairwise orthogonal distributions. We define the mixed scalar curvature of this structure and prove integral formulas generalizing classical and recent results on foliations and distributions generating the tangent bundle of a manifold. Examples with one-dimensional distributions, paracontact manifolds, hypersurfaces in space forms, etc., illustrate the results. [ABSTRACT FROM AUTHOR]

We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the canonical variation of the metric. [ABSTRACT FROM AUTHOR]

We introduce a notion of equivalence for singular foliations—understood as suitable families of vector fields—that preserves their transverse geometry. Associated with every singular foliation, there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence, we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980s. [ABSTRACT FROM AUTHOR]

In this paper we investigate the spectral sequence associated with a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples. [ABSTRACT FROM AUTHOR]

In an earlier paper, we studied manifolds M endowed with a generalized F structure $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ , skew-symmetric with respect to the pairing metric, such that $$\Phi ^3+\Phi =0$$ . Furthermore, if $$\Phi $$ is integrable (in some well-defined sense), $$\Phi $$ is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ , where $$A^3+A=0$$ and some relations between A and $$\pi $$ hold. We establish the integrability conditions in terms of $$(A,\pi )$$ . They include the facts that A is a classical CRF structure, $$\pi $$ is a Poisson bivector field and $$\mathrm{im}\,A$$ is a (non)holonomic Poisson submanifold of $$(M,\pi )$$ . We discuss the case where either $$\mathrm{ker}\,A$$ or $$\mathrm{im}\,A$$ is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of $$\mathrm{im}\,A$$ inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of $$\pi $$ , including an associated spectral sequence and a Dolbeault type grading. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle. [ABSTRACT FROM AUTHOR]

A singular Riemannian foliation $${\mathcal{F}}$$ on a complete Riemannian manifold M is called a polar foliation if, for each regular point p, there is an immersed submanifold Σ, called section, that passes through p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of M into the orbits of a polar action, i.e., an isometric action with sections. In this article we prove that the leaves of $${\mathcal{F}}$$ coincide with the level sets of a smooth map H: M → Σ, if M is simply connected. In particular, the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng. [ABSTRACT FROM AUTHOR]

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kähler-Einstein manifolds or over coadjoint orbits of a compactly connected semisimple Lie group. [ABSTRACT FROM AUTHOR]

We consider planar vector fields without zeroes ξ and study the image of the associated Lie derivative operators L acting on the space of smooth functions. We show that the cokernel of L is infinite-dimensional as soon as ξ is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of ξ is 'simple enough' (e.g. if ξ is polynomial) then ξ is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of L and to build an embedding of $${\mathbb{R}^{2}}$$ into $${\mathbb{R}^{4}}$$ which rectifies ξ. Finally, we use this embedding to characterize the functions in the image of L. [ABSTRACT FROM AUTHOR]

We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms. [ABSTRACT FROM AUTHOR]

Let $${(M,\mathcal{F})}$$ be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino’s commuting sheaf of $${(M,\mathcal{F})}$$ vanish if $${(M,\mathcal{F})}$$ is developable and π1 M is of polynomial growth. By theorems of Álvarez López in (Álvarez López, Ann. Global Anal. Geom., 10:179–194, 1992) and (Álvarez López, Ann. Pol. Math., 64:253–265, 1996), our result implies that $${(M,\mathcal{F})}$$ is minimizable under the same conditions. As a corollary, we show that $${(M,\mathcal{F})}$$ is minimizable if $${\mathcal{F}}$$ is of codimension 2 and π1 M is of polynomial growth. [ABSTRACT FROM AUTHOR]

We study the existence and stability of holomorphic foliations in dimension greater than 4 under perturbations of the underlying almost-complex structure. An example is given to show that, unlike in dimension 4, J-holomorphic foliations are not stable under large perturbations of almost-complex structure. [ABSTRACT FROM AUTHOR]

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature. [ABSTRACT FROM AUTHOR]

Abstract  New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the harmonic unit vector field tangent to a parallel family of hyperbolic geodesics is shown to be unstable, by constructing a class of compactly supported energy-decreasing variations. All examples considered have infinite total bending. [ABSTRACT FROM AUTHOR]

Abstract  A contamination in a 3-manifold is an object interpolating between the contact structure and the lamination. Contaminations seem to provide a link between 3-dimensional contact geometry and the classical topology of 3-manifolds, as described in a separate paper (Oertel and Świa̧tkowski, Contact structures, σ-confoliations, and contaminations in 3-manifolds. arXiv math.GT/0307177). In this article we deal with contaminations carried by branched surfaces, giving a sufficient condition for a branched surface to carry a pure contamination. [ABSTRACT FROM AUTHOR]

Abstract  We obtain a vanishing theorem for the half-kernel of a transverse Spin c Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a sufficiently large power of a line bundle, whose curvature vanishes along the leaves and is transversely non-degenerate at any point of the ambient manifold. [ABSTRACT FROM AUTHOR]

Abstract  Using a relation between the terms of the spectral sequence of a Riemannian foliation and its adiabatic limit, we obtain Bochner type techniques for this special setting and, as a consequence, in the special case of a Riemannian flow we obtain vanishing conditions for the top dimensional group of the basic cohomology $$H_{b}^{q}(\mathcal{F})$$-which is related to the property of being geodesible. We also extend a Weitzenbck type formula for the leafwise Laplacian and, for the particular class of compact foliations, we obtain a generalization of a result due to Ph. Tondeur, M. Min-Oo, and E. Ruh concerning the vanishing of the basic cohomology under the assumption that certain curvature operators are positive definite. In the final part we present an example. [ABSTRACT FROM AUTHOR]