Pullback Bundle Research Articles

Pull-back vector fields' bienergy

This work delves into the bienergy of a pull-back vector field from a Riemannian manifold (M, g) to its tangent bundle TN, which is endowed with the Sasaki metric hs. The central question under investigation is whether such a pull-back vector field exhibits biharmonic properties. Through rigorous proof, we establish that if is a parallel vector field, then the pull-back bundle V ∊ Γ(φ−1(TN)) indeed manifests as biharmonic. This finding is significant, particularly for pull-back vector fields on compact manifolds (M, g) that encompass the harmonic map φ. The implications of this work contribute to a deeper understanding of the interactions between vector fields and harmonic mappings within the geometry of Riemannian manifolds, potentially paving the way for further explorations in the realm of differential geometry and its applications. The results affirm the biharmonic nature of parallel vector fields in the specified context, thereby enriching the theoretical framework surrounding the study of bienergy and harmonic maps.

Pullback Bundles and the Geometry of Learning.

Explainable Artificial Intelligence (XAI) and acceptable artificial intelligence are active topics of research in machine learning. For critical applications, being able to prove or at least to ensure with a high probability the correctness of algorithms is of utmost importance. In practice, however, few theoretical tools are known that can be used for this purpose. Using the Fisher Information Metric (FIM) on the output space yields interesting indicators in both the input and parameter spaces, but the underlying geometry is not yet fully understood. In this work, an approach based on the pullback bundle, a well-known trick for describing bundle morphisms, is introduced and applied to the encoder-decoder block. With constant rank hypothesis on the derivative of the network with respect to its inputs, a description of its behavior is obtained. Further generalization is gained through the introduction of the pullback generalized bundle that takes into account the sensitivity with respect to weights.

Weighted $$L^2$$ Holomorphic Functions on Ball-Fiber Bundles Over Compact Kähler Manifolds

Let $${\widetilde{M}}$$ be a complex manifold, $$\Gamma $$ be a torsion-free cocompact lattice of $$\text {Aut}({\widetilde{M}})$$ and $$\rho :\Gamma \rightarrow SU(N,1)$$ be a representation. Suppose that there exists a $$\rho $$ -equivariant totally geodesic isometric holomorphic embedding $$\imath :{{\widetilde{M}}}\rightarrow {\mathbb {B}}^N$$ . Let $$M:={{\widetilde{M}}}/\Gamma $$ and $$\Sigma :={\mathbb {B}}^N/\rho (\Gamma )$$ . In this paper, we investigate a relation between weighted $$L^2$$ holomorphic functions on the fiber bundle $$\Omega :=M\times _\rho {\mathbb {B}}^N$$ and the holomorphic sections of the pull-back bundle $$\imath ^*(S^mT^*_\Sigma )$$ over M. In particular, $$A^2_\alpha (\Omega )$$ has infinite dimension for any $$\alpha >-1$$ and if $$n<N$$ , then $$A^2_{-1}(\Omega )$$ also has the same property. As an application, if $$\Gamma $$ is a torsion-free cocompact lattice in SU(n, 1), $$n\ge 2$$ , and $$\rho :\Gamma \rightarrow SU(N,1)$$ is a maximal representation, then for any $$\alpha >-1$$ , $$A^2_\alpha ({\mathbb {B}}^n\times _{\rho } {\mathbb {B}}^N)$$ has infinite dimension. If $$n<N$$ , then $$A_{-1}^2({\mathbb {B}}^n\times _{\rho } \mathbb B^N)$$ also has the same property.

Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

Uniqueness of 1D Generalized Bi-Schrödinger Flow

We consider the initial value problem for the generalized bi-Schrödinger flow equation for maps from the one-dimensional flat torus into a compact locally Hermitian symmetric space. The governing equation, which is satisfied by sections of the pull-back bundle induced from the flow, is a fourth-order extension of the 1D Schrödinger map flow equation with loss of derivatives. As the author’s previous work, time-local existence of a smooth solution has been obtained by an intrinsic approach based on the geometric energy method. In the present paper, as a continuation of the work, we establish the uniqueness of the solution. To show the uniqueness, we adopt an extrinsic approach to compare two solutions via an isometric embedding into an ambient Euclidean space. We introduce an energy modifying the classical \(H^2\)-energy for the difference of two solutions, which enables us to eliminate the difficulty of the loss of derivatives. In order to achieve this, we establish some useful tools of computation exploiting the geometric structure of the locally Hermitian symmetric space in the ambient Euclidean space.

Horizontal lift problems in a pull-back bundle of tensor bundles defined by projection of the cotangent bundles

Using projection (submersion) of the cotangent bundle T*M over a manifold M, we define a semi-tensor (pull-back) bundle tM of type (p,q). The purpose of this paper is to investigate horizontal lifts of vector fields in a special class of semi-tensor (pull-back) bundle tM of type (p,q).

On Pull-Back Bundle of Tensor Bundles Defined by Projection of The Cotangent Bundle

Abstract Using projection (submersion) of the cotangent bundle T*M over a manifold M, we define a semi-tensor (pull-back) bundle tM of type (p,q). The aim of this study is to investigate complete lift of vector fields in a special class of semi-tensor bundle tM of the type (p,q). We also have a new example for good square in this work.

Spherical harmonic d-tensors

Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group [Formula: see text]. The aim of this work is to make use of this tool also in the setting of Finsler geometry, or more general geometries on the tangent bundle, where the objects of relevance are d-tensors on the tangent bundle, or tensors in a pullback bundle, instead of ordinary tensors. For this purpose, we construct a set of d-tensor harmonics for spherical symmetry and show how these can be used for calculations in Finsler geometry.

Hermitian Yang–Mills Connections on Blowups

Consider a vector bundle over a Kahler manifold which admits a Hermitian Yang–Mills connection. We show that the pullback bundle on the blowup of the Kahler manifold at a collection of points also admits a Hermitian Yang–Mills connection, for Kahler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explicit. This recovers, through the Hitchin–Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.

On Holomorphic Curvature of Complex Finsler Square Metric

The notion of the holomorphic curvature for a Complex Finsler space (M,F) is defined with respect to the Chern complex linear connection on the pull-back tangent bundle. This paper is about the fundamental metric tensor, inverse tensor and as a special approach of the pull-back bundle is devoted to obtaining the holomorphic curvature of Complex Finsler Square metrics. Further, it proved that it is not a weakly Kahler.

Diagonal lift in the semi-cotangent bundle and its applications

The present paper is devoted to some results concerning the diagonal lift of tensor fields of type (1,1) from manifold M to its semi-cotangent bundle t*M. In this context, cross-sections in the semi-cotangent (pull-back) bundle t*M of cotangent bundle T*M by using projection (submersion) of the tangent bundle TM can be also defined.

On Semi-Tensor Bundle

We investigate some lifts of tensor fields of type (1,0) on a cross-section in the semi-tensor (pullback) bundle tM of tensor bundle TM of type (p,q) by using projection (submersion) of the cotangent bundle T*M and we find some relation for them.

Mathematical fixation: Search viewed through a cognitive lens.

We provide a mathematical category theory account of the size and location of the authors' Functional View Field (FVF). Category theory explains systematic cognitive ability via universal construction, that is, a necessary and sufficient condition for composition of cognitive processes. Similarly, FVF size and location is derived from a (universal) construction called a fibre (pullback) bundle.

Strict orbifold atlases and weighted branched manifolds

This note revisits the ideas in an earlier (2007) paper on orbifolds and branched manifolds, showing how the constructions can be simplified by using a version of the Kuranishi atlases recently developed by McDuff--Wehrheim. We first show that every orbifold has such an atlas, and then use it to obtain explicit models first for the nonsingular resolution of an oriented orbifold (which is a weighted nonsingular groupoid with the same fundamental class) and second for the Euler class of an oriented orbibundle. In this approach, instead of appearing as the zero set of a multivalued section, the Euler class is the zero set of a single-valued section of the pullback bundle over the resolution, and hence has the structure of a weighted branched manifold in which the weights and branching are canonically defined by the atlas.

Characterization of exact lumpability for vector fields on smooth manifolds

We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from the literature that exist for Euclidean spaces. We introduce a partial connection on the pullback bundle that is related to the Bott connection and behaves like a Lie derivative. The lumping conditions are formulated in terms of the differential of the lumping map, its covariant derivative with respect to the connection and their respective kernels. Some examples are discussed to illustrate the theory.

A Fourth-Order Dispersive Flow into Kähler Manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.

Generalized Sasaki Metrics on Tangent Bundles

Abstract We define a class of metrics that extend the Sasaki metric of the tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback bundle π−1(TM⊕T*M), where π : T M → M is the natural projection. We obtain the expression of the transferred metric and define a canonical metric connection with torsion. We calculate the torsion, curvature and Ricci curvature of this connection and give a few applications of the results. We also discuss the transfer of generalized complex and generalized Kähler structures from the pullback bundle to the tangent manifold.

An unconstrained Lagrangian formulation and conservation laws for the Schrödinger map system

We consider energy-critical Schrödinger maps from \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^2$\end{document}R2 into \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^2$\end{document}S2 and \documentclass[12pt]{minimal}\begin{document}$\mathbb {H}^2$\end{document}H2. Viewing such maps with respect to orthonormal frames on the pullback bundle provides a gauge field formulation of the evolution. We show that this gauge field system is the set of Euler-Lagrange equations corresponding to an action that includes a Chern-Simons term. We also introduce the stress-energy tensor and derive conservation laws. In conclusion we offer comparisons between Schrödinger maps and the closely related Chern-Simons-Schrödinger system.

LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

Abstract. Let M be a complete Riemannian manifold and let N be aRiemannian manifold of non-positive sectional curvature. Assume thatRic M ≥ − 4(p−1)p 2 µ 0 at all x ∈ M and Vol(M) is infinite, where µ 0 > 0 isthe infimum of the spectrum of the Laplacian acting on L 2 -functions onM. Then any p-harmonic map φ : M → N of finite p-energy is constant.Also, we study Liouville type theorem for p-harmonic morphism. 1. IntroductionLet (M,g) and (N,h) be smooth Riemannian manifolds and let φ : M → Nbe a smooth map. For a compact domain Ω ⊂ M, the p-energy E p (φ;Ω) of φover Ω is defined by(1.1) E p (φ;Ω) =1pZ Ω |dφ| p µ M ,where the differential dφ is a section of the bundle T ∗ M ⊗ φ −1 TN → M andφ −1 TN denotes the pull-back bundle via the map φ. The bundle T ∗ M ⊗φ −1 TN → M carries the connection ∇ induced by the Levi-Civita connectionson M and N. A map φ : M → N is called p-harmonic if the p-tension fieldτ p (φ) = 0, which is defined by(1.2) τ p (φ) = tr g ∇(|dφ| p−2 dφ),where tr g denote the trace with respect to the metric g. A p-harmonic map φ isa critical point of the energy functional defined by (1.1) on any compact domainΩ ⊂ M. When p = 2, p-harmonic maps are well-known to be harmonic maps.Several studies are given for harmonic maps (see [5], [6], [7], [8], [10], [11], [12],[13], [14], [16]). Let µ

Semi-cotangent bundle and problems of lifts

Using the ber bundle M over a manifold B, we dene a semi-cotangent (pull-back) bundle t B, which has a degenerate symplectic structure. We consider lifting problem of projectable geometric objects on M to the semi-cotangent bundle. Relations between lifted objects and a degenerate symplectic structure are also presented.