Tangent Bundle Research Articles

Linearization of nonlinear Fokker-Planck equations and applications

Let P be the space of probability measures on Rd. We associate a coupled nonlinear Fokker-Planck equation on Rd, i.e. with solution paths in P, to a linear Fokker-Planck equation for probability measures on the product space Rd×P, i.e. with solution paths in P(Rd×P). We explicitly determine the corresponding linear Kolmogorov operator L˜t using the natural tangent bundle over P with corresponding gradient operator ∇P. Then it is proved that the diffusion process generated by L˜t on Rd×P is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by L˜t in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurability conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of “Nemytskii-type”. As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schrödinger type PDEs on Rd×P2, through the Feynman-Kac formula for the corresponding diffusion processes.

Characteristic classes for families of bundles

The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying tau _M-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for tau _M-fibrations, Baut(tau _M), and its cohomology ring, i.e., the ring of characteristic classes of tau _M-fibrations. For a bundle xi over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal xi -fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of Baut(xi ) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given tau _M-fibration comes from a manifold bundle.

Examples of Fano manifolds with non‐pseudoeffective tangent bundle

Let X $X$ be a Fano manifold. While the properties of the anticanonical divisor − K X $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle T X $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of T X $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.

Stability of the tangent bundles of complete intersections and effective restriction

For n≥3 and r≥1, let M be an (n+r)-dimensional irreducible Hermitian symmetric space of compact type and let 𝒪 M (1) be the ample generator of pic(M). Let Y=H 1 ∩⋯∩H r be a smooth complete intersection of dimension n, where H i ∈|𝒪 M (d i )| with d i ≥2. We prove a vanishing theorem for twisted holomorphic forms on Y. As an application, we show that the tangent bundle T Y of Y is stable. Moreover, if X is a smooth hypersurface of degree d in Y such that the restriction pic(Y)→pic(X) is surjective, we establish some effective results for d to guarantee the stability of the restriction T Y | X . In particular, if Y is a general hypersurface in ℙ n+1 and X is a general smooth divisor in Y, we show that T Y | X is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.

Kenmotsu manifolds endowed with the semi-symmetric non-metriC $\phi$-connection to its tangent bundle

Kenmotsu manifolds endowed with the semi-symmetric non-metriC $\phi$-connection to its tangent bundle

The singular evolutoids set and the extended evolutoids front

In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most cusp singularities. By the Gauss-Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary (Theorem 2.20 in [4]) applied to the extended front of evolutoids of a hedgehog we obtain an integral equality for smooth periodic curves.

Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor

We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties byQ\mathbb {Q}-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

AbstractLetXbe a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus ofX: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale coverXsplits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf ofXaccording to its holonomy representation. In particular, we classify thoseXwhich have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi–Yau or irreducible holomorphic symplectic. As an application of these results, we show that ifXhas dimension four, then it satisfies Campana’s Abelianity Conjecture.

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

This paper is devoted to the study of the relation between `formal exponential maps,' the Atiyah class, and Kapranov $L_\infty[1]$ algebras associated with dg manifolds in the $C^\infty$ context. Given a dg manifold, we prove that a `formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an $L_\infty[1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ arising from a complex manifold $X$, we prove that this $L_\infty[1]$ algebra structure is quasi-isomorphic to the standard $L_\infty[1]$ algebra structure on the Dolbeault complex $\Omega^{0,\bullet}(T^{1,0}_X)$.

Studying finite-time (non)-domination in dynamical systems using Oseledec’s splitting. Application to the standard map

Oseledec’s theorem provides the necessary conditions for the existence of the decomposition of the tangent bundle called Oseledec’s splitting and the formal definition of the Lyapunov exponents. Using the concept of τ-domination of expansion or contraction rates of invariant subspaces after a time τ, together with the Oseledec theorem, we propose three quantifiers of τ-domination and τ-non-domination, which can be used to study the complex dynamics of physical systems. The interpretation of the Oseledec τ-domination in terms of the finite-time Lyapunov exponents is of great advantage for this purpose. Numerical results for the quantifiers are presented using the conservative standard map in a regime with mixed phase-space dynamics. Distinct typical regions in the phase-space are chosen, two containing a completely chaotic motion and two containing regularity islands. Results show that all quantifiers recognize that regions close to hyperbolic periodic points have a larger Oseledec τ-domination than regions around islands of regularity. Furthermore, Oseledec τ-non-dominated regions are also identified and are mostly close to regularity islands. We found a relation between the mean of expansion rates between the Oseledec subspaces and the mean of the angles between them.

Derivatives with respect to horizontal and vertical lifts of the deformed complete lift metric G_{f} on tangent bundle.

In this paper, we define the deformed complete lift metric G_{f} on tangent bundle, which is completely determined by its action on vector fields of type X^{H} and ω^{V}. Later, we obtain the covarient and Lie derivatives applied to the deformed complete lift metric G_{f} with respect to the horizontal and vertical lifts of vector fields, respectively.

On unit sphere tangent bundles over complex Grassmannians

On unit sphere tangent bundles over complex Grassmannians

Complex Functional Maps: A Conformal Link Between Tangent Bundles

AbstractIn this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is theirorientation awareness. More specifically, we demonstrate that unlike regular functional maps that linkfunctional spacesof two manifolds, our complex functional maps establish a link betweenoriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting operations become naturally orientation‐aware, thus favouringorientation and angle preserving correspondenceacross shapes, without relying on descriptors or extra regularization. Finally, and perhaps more importantly, we demonstrate how these objects enable several practical applications within the functional map framework. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation, regularizing pipelines that previously suffered from orientation‐reversing symmetry errors.

Evolution of Nonlinear Reduced-Order Solutions for PDEs with Conserved Quantities

Reduced-order models of time-dependent partial differential equations (PDEs) where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time varying parameters have thus far been derived in an ad hoc manner. Here, we introduce reduced-order nonlinear solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent parameters. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. This results in a set of explicit ordinary differential equations on the tangent bundle of the manifold. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems. We demonstrate the efficacy of RONS on three examples: an advection-diffusion equation, the nonlinear Schrödinger equation, and Euler's equation for ideal fluids.

Distribution in the unit tangent bundle of the geodesics of given type

AbstractRecall that two geodesics in a negatively curved surface S are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure ${\mathfrak {m}}^S$ on $T^1S$ . We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups

Abstract We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in PSL 2 ⁢ ( ℝ ) {{\mathrm{PSL}_{2}(\mathbb{R})}} . We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.

Ricci soliton on the tangent bundle with semi-symmetric metric connection

In this paper, we studied the tangent bundle endowed with semi-symmetric metric connection obtained by vertical and complete lifts of a semi-symmetric metric P-connection on the base manifold. Firstly, we give a relationships between (TM; gc) and (M; g) to be an Einstein manifolds. Secondly, we investigate necessary and su cient conditions for (TM; gc) with complete and vertical lift of torqued potential elds to be Ricci soliton.

On the Biharmonicity of Vector Fields and Unit Vector Fields

Let (M, g) be a compact Riemannian manifold. Equipping its tangent bundle TM (resp. unit tangent bundle \(T_1M\)) with a pseudo-Riemannian g-natural metric G (resp. \({\tilde{G}}\)), we study the biharmonicity of vector fields (resp. unit vector fields) as maps \((M,g) \rightarrow (TM,G)\) (resp. \((M,g) \rightarrow (T_1M,{\tilde{G}})\)), as well as critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) of vector fields (resp. unit tangent bundles) on M. Contrary to the Sasaki metric on TM, where the two notions are equivalent to the harmonicity of the vector field and then to its parallelism, we prove that for large classes of g-natural metrics on TM, the two notions are not equivalent. Furthermore, we give examples of vector fields which are biharmonic as critical points of the bienergy functional restricted to \({\mathfrak {X}}(M)\), but are not biharmonic maps. We provide equally examples of proper biharmonic vector fields (resp. unit vector fields), i.e. those which are critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) without being harmonic.

Toric foliations with split tangent sheaf

We study holomorphic foliations of arbitrary codimension in smooth complete toric varieties. We show that split foliations are stable if some good behaviour of their singular set is provided. As an application of these results, we exhibit irreducible components of the space of foliations that arise as pullbacks of some special T-invariant subvarieties.