Natural Foliation Research Articles

Sasaki–Randers metric in Finsler geometry

In this paper, we consider the Randers metric F = α + β on a smooth manifold M and by using it define Sasaki–Randers metric G on the slit tangent bundle TM ∘ . We introduce natural foliations on ( TM ∘ , G ) and study totally geodesic and bundle-likeness properties of them. Then, we construct a framed f ( 3 , − 1 ) -structure on TM ∘ and show that it reduces to an almost paracontact structure on the indicatrix bundle.

The warped Sasaki–Matsumoto metric and bundlelike condition

Let \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^n=(M,F)$\end{document}Fn=(M,F) be a Finsler manifold and G be the Sasaki–Matsumoto metric on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘. Bejancu and Farran [“Finsler geometry and natural foliations on the tangent bundle,” Rep. Math. Phys. 58, 131 (2006)] proved that \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^n=(M,F)$\end{document}Fn=(M,F) is a Riemannian manifold if and only if the Sasaki–Matsumoto metric G on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘ is bundlelike for the vertical foliation. Let \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_1+n_2}=(M_1\times _fM_2,F)$\end{document}Fn1+n2=(M1×fM2,F) be the warped product Finsler manifold. In this paper the warped Sasaki–Matsumoto metric \documentclass[12pt]{minimal}\begin{document}${}^*\mathbf {G}$\end{document}*G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then \documentclass[12pt]{minimal}\begin{document}${}^*\mathbf {G}$\end{document}*G on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘ is bundlelike for the warped vertical foliation \documentclass[12pt]{minimal}\begin{document}$\mathcal {V}^*(TM^\circ )$\end{document}V*(TM∘) if and only if \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_1}_1=(M_1,F_1)$\end{document}F1n1=(M1,F1) and \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_2}_2=(M_2,F_2)$\end{document}F2n2=(M2,F2) are Riemannian manifolds.

From TLS to VLS: Biomass Estimation at Individual Tree Level

This study explores the applicability of vehicle-based laser scanning (VLS) for biomass estimation at individual tree level, since biomass serves as an essential biophysical parameter indicating tree health. Previous work suggests that terrestrial laser scanning (TLS) has been primarily validated for biomass prediction, however, in subject to laborious relocation in practice. VLS, as an advanced mode of TLS with more flexible mobility and also high sampling density, can work as a new efficient technique for surveying single trees. Combined with the positive binds between the biomass and TLS-samplings during manual defoliation, this work aims to seek the relations between biomass and VLS-samplings, by correlating the VLS- and TLS-samplings within the same crowns during natural foliation. The resulting R2 values of the two correlations after normalization are larger than 0.88 and 0.61, respectively, and the associated root mean square errors (RMSEs) are less than 0.051 and 0.076. VLS, thus, can be validated for estimating biomass at the individual tree level, with the TLS-investigated data as a bridging reference.

Gravitational instantons, self-duality, and geometric flows

We discuss four-dimensional "spatially homogeneous" gravitational instantons. These are self-dual solutions of Euclidean vacuum Einstein's equations with potentially non-vanishing cosmological constant. They are endowed with a product structure R \times M_3 leading to a natural foliation into three-dimensional subspaces evolving in Euclidean time. For a large class of three-dimensional subspaces, the dynamics coincides with the geometric flow on the three-dimensional homogeneous slice, driven by the Ricci tensor plus an so(3) gauge connection. The metric on the three-dimensional space is related to the vielbein of the three-dimensional subspace, while the gauge field is inherited from the anti-self-dual component of the four-dimensional Levi--Civita connection.

The automorphism group of the tetrablock

The tetrablock is a domain in 3-dimensional complex space that meets 3-dimensional Euclidean space in a regular tetrahedron. It is shown to be inhomogeneous and its automorphism group is determined. A type of Schwarz lemma for the tetrablock is proved. The action of the automorphism group is described in terms of a certain natural foliation of the tetrablock by complex geodesic discs.

Deformation of holomorphic maps onto the blow-up of the projective plane

Let S be the blow-up of the projective plane at d distinct points and ψ : S ′ → S be any surjective holomorphic map from a compact complex manifold S ′ . We will show that all deformations of ψ come from automorphisms of S if d ⩾ 3 . The result is optimal in the sense that it is not true if d ⩽ 2 . The strategy of the proof is to use the infinitesimal automorphisms of the web geometry on S arising from the natural foliations of S induced by the pencils of the lines through the blow-up centers.

Finsler geometry and natural foliations on the tangent bundle

A new approach for the study of geometry of Finsler manifolds is proposed. Let F 'n = (M, F) he an m-dimensional Finsler manifold and G be the Sasaki-Finsler metric on TM ° = TM\{0}. We exploit many natural foliations of (TM °, G) by showing that their geometry is closely related to that of the Finsler manifold itself. In particular, we use this approach to obtain new characterizations of Finsler manifolds with positive constant curvature.

Generalized immersions and the rank of the second fundamental form

Partial differential equations and differential geometry come together in the idea of a generalized immersion. This concept, defined by means of Grassmann bundles and contact forms, allows for ?immersions? with ?singularities.? Sophus Lie?s generalized solutions to partial differential equations are an important special case. The classical second fundamental form has a natural generalization in the context of generalized immersions. The rank of the form is then meaningful. A constant rank assumption on the generalized second fundamental form leads to a natural foliation of the generalized immersion, at least when the ambient space is a space of constant curvature. Questions about the total geodesy and regularity of the foliation are also addressed.

Hilbert modular foliations on the projective plane

We describe explicitly holomorphic singular foliations on the projective plane corresponding to natural foliations of Hilbert modular surfaces associated to the field Q. These are concrete models for a very special class of foliations in the recent birational classification of foliations on projective surfaces.

On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows

The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x, t), h(x, t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in Part 1 of this series.Finally, the general stability criterion is applied to a variety of classes of steady states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

Building a Quantum Spacetime

A quantum spacetime is constructed from the freedata given at past null infinity. One starts with afield equation for a scalar function Z on the initialsurface and then shows that the solution depends on four constants of integration. Theseconstants become the spacetime points and the levelsurfaces of the scalar function, i.e., Z = const, becomenull hypersurfaces on the derived spacetime. A phase space together with a complex structure areconstructed on past null infinity. This Hilbert space ofincoming gravitons possesses a natural foliation whichdefines superselection sectors on the space of asymptotic quantum states. The dynamics of nullsurface quantization provides spacetime-valued quantumoperators on the superselection sectors. It is shownthat the spacetime points themselves become operators with nonvanishing commutationrelations.

On tori without conjugate points

In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.

On the characteristic classes of actions of lattices in higher rank Lie groups

We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups. Let F be a discrete group acting by diffeomorphisms on a smooth compact manifold M. Associated to this action are certain characteristic classes in H (F, R), which are constructed as characteristic classes of a natural foliation associated to the group action. The action of F on M induces an action of F on the principal frame bundle P(M) of M and the characteristic classes of the action can be interpreted as obstructions to the existence of invariant geometric structures on M, i.e., principal subbundles of P(M) invariant by the F-action. If F is a lattice in a higher rank semisimple Lie group, then F has strong rigidity properties (see, e.g., [M, Z1]). In an earlier paper [S], we showed, using techniques from ergodic theory, that for a certain class of F-actions there is always an invariant measurable reductive geometric structure, i.e., a measurable principal subbundle with reductive structure group, which is invariant by the F-action. Moreover, the noncompact semisimple part of this reductive group is locally isomorphic to a semisimple factor of the ambient Lie group of F [Zi]. Zimmer [Z3] recently proved this result for a large class of actions (which does not a priori include the class of actions considered in [S]). A natural question is whether these results remain true in the smooth category. The purpose of this paper is to show that the characteristic classes, which obstruct a smooth geometric structure, vanish in the presence of a measurable geometric structure. Explicitly, we have Main Theorem. Let (M, Y) be a codimension n, C 2-foliated manifold and suppose that the linear holonomy cocycle is measurably equivalent to a locally tempered cocycle taking values in a subgroup H c GL(n, R) which is stable under transpose. Then the Weil homomorphism X: H'(g[(n), 0(n)) -Hc(M, Yi) Received by the editors February 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R32; Secondary 57S20. Research partially supported by an Alfred P. Sloan Dissertation Fellowship. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Qualitative study of motion under the potentials

The two-parameter family of hamiltonians systems defined by $$H_{(a,\alpha )} (r,\theta ,p_r ,p_\theta ) = (p_r^2 + p_\theta ^2 r^{ - 2} )/2 - ar^{ - \alpha } ,\alpha ,a \in \Re ^ +$$ is considerated. For every value of $$(a,\alpha ) \in \Re ^ + \times \Re ^ + ,H_{(a,\alpha )}$$ is integrable, the energy H(a,α) and the angular momentum C(r,θ,pr,pθ)=pθ being two independent first integrals. Using appropriate coordinates, the natural foliation ofI h = ∪ c I hc is found-up to now, the sets Ih∩{c≥0} and Ih∩{c≤0} were studied separately and then their boundaries identified ([CLL], [D], [LL], for instance). As a corollary to this topological description, we obtain a relation between the value of the parametre α and the variation of the θ-coordinate along an orbit while it stays near collision or infinity.

Killing vectors and maximal slicing in general relativity

In solutions to Einstein's equations, killing vectors and maximal slices are locked together. If an asymptotically flat spacetime has a maximal slice and a killing vector which lies in the maximal slice at infinity, then the killing vector lies in the slice everywhere. For closed, compact, without boundary solutions one has a more general result. In this case if one has a slice with the trace of the extrinsic curvature constant (not necessarily zero) then all killing vectors must lie in the slice. The only exception is closed flat space. This means that maximal slices form a natural foliation of any spacetime with a symmetry.

Independent Variation of Secondary Classes

teristic classes of a certain class of foliations. The examples we compute show that many of these secondary classes vary linearly independently. This generalizes a result due to Thurston [T] on the variation of one of these classes, the Godbillon-Vey class. It is also a partial analogue of the results of Bott [B2] and Lazarov-Pasternack [LP] on the independent variation of the secondary classes for holomorphic and Riemannian foliations respectively. We are also able to compute the Simons' characters of these foliations and we show that many of them vary linearly independently. The method we use is a generalization of the theory of residues of singular foliations due to Baum-Bott [BB]. We work with the natural foliation z on a flat vector bundle and a vector field X tangent to the fiber of the bundle which preserves the foliation. We assume the vector field has an isolated singularity along the zero section. This situation determines certain cohomology classes on the zero section, the residues of z- and X. We relate these residues to the secondary characteristic classes and Simons' characters of the foliation ' spanned by z and X off the zero section. Using this technique we compute examples which show that the residues of Z and X and thus the characteristic classes of Z vary linearly independently. In Section 2 we record some facts we will use later. In Section 3 we

Development of axial-plane slaty cleavage and schistosity in experimental and natural materials

An experiment is described in which folds and an axial-plane foliation were produced in an artificial layered specimen made from salt and mica. The foliation differs on a micro-scale from natural foliations due to the unnatural experimental conditions. Taking results of annealing experiments into account a model is presented for the development of natural foliations. The model involves kinking of an earlier foliation and later or concurrent migration of kink-band boundaries and growth of new micas parallel to old micas and to kink-band boundaries. Four examples of natural schists are described and the possibility of the mechanism operating in the development of axial-plane slaty cleavage and schistosity is discussed. It is concluded that the mechanism probably operates in both cases but alone it cannot explain the domainal fabric commonly found in slates.

Propagation of Analytic and Differentlable Singularities for Solutions of Partial Differential Equations

In the first part of this paper, we study propagation of singularities for solutions of an analytic pseudo-differential equation, the characteristic set of which is a regular involutive manifold. There exists a natural foliation of this manifold, and (theorem 1.7) the analytic singular spectrum of a solution is a union of leaves—this is a joint work P. Schapira. The same result holds for differentiable singular spectrum (wave front) assuming Levi's condition. A similar result had been proved by J. Sjostrand for C°° pseudo-differential equations, but with additional assumptions on complex characteristics [14]. We prove also microlocal solvability for our operators (theorem 1.6). Complete proofs are given in [6] and [4]. In the second part of this paper (§ 4 to 7), we study operators the characteristic set of which contains a regular involutive manifold. The main result (theorem 5.6) is an analogue, for singular spectrum, to Holmgren theorem for support. Applications to propagation of analytic singularities and to uniqueness of Cauchy problem are given. Besides arguments used in the first part, we use results of Kashiwara [11], [12] and direct infinitesimal geometry. A more detailed exposition is given in [5].