Carlos A. Torre
We define a subcategory of the category of diffeological spaces, which contains smooth manifolds, the diffeomorphism subgroups and its coadjoint orbits. In these spaces we construct a tangent bundle, vector fields and a de Rham cohomology.
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We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle $T^H X$ defined by Hector is a diffeological vector space over $X$ when $X$ is…
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and its expected properties, we turn to our main interest, which is constructing pseudo-bundles of diffeological Clifford algebras and those of diffeological Clifford modules, by means of the procedure called diffeolo…
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. We prove fundamental results about these tangent spaces, compute them in many examples, and observe that while they agree for smooth manifolds and many of the exam…
We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections. One aspect prominent in the diffeological context has to do with the choice of an appropriate substitute for tangent vectors and smooth vector fields, since there are not yet standard counterparts for these notions. In this respect we opt for the simplest possibility; since the…
On diffeological principal bundles of non-formal pseudo-differential operators over formal ones
本文研究非形式经典伪微分算子相对于形式算子的微分拓扑主丛及其结构群，借助无局部平凡化主丛结果与既有光滑化联络加以刻画，并提出开放问题。
On the geometry of diffeological vector pseudo-bundles and infinite dimensional vector bundles: automorphisms, connections and covariant derivatives
本文研究微分拓扑向量伪丛范畴，推广经典微分几何工具至该框架，揭示联络1-形式与协变导数不同构等差异，应用于带奇点有限维例子及无限维向量丛。
A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual o…
We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring …
Infinitely Many Tangent Functors on Diffeological Spaces
本文研究 diffeological 空间上的切 functor，构建了无穷多个非同构的切 functor，并表明在非光滑流形外，切 functor 的选择并不唯一。
We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. Our bijections are established by explicit group isomorphisms: transgression and regression. Restricted to smooth, finite-dimensional manifolds, our results extend previous work of J. W. Barrett.
Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in arXiv:1509.03023. We give more details regarding the behavior of this operation with respect to gluing, also providing some details omitted from arXiv:1509.03023, and pay more attention to the relations with the spaces of smooth maps. We also show that a usual smooth vector bundle over a manifold that admits a finite atlas can be seen as a result of a d…
The Structure of the Internal Tangent Space to a Point of the Orbit Space of a Manifold under a Proper Lie Group Action
本文研究真李群作用下流形轨道空间作为微分空间时一点处的内切空间，给出其一般描述，并证明其与轨道型分层意义下的分层切空间同构。
We introduce a new notion of Morita equivalence for diffeological groupoids, generalising the original notion for Lie groupoids. For this we develop a theory of diffeological groupoid actions, -bundles and -bibundles. We define a notion of principality for these bundles, which uses the notion of a subduction, generalising the notion of a Lie group(oid) principal bundle. We say two diffeological groupoids are Morita equivalent if and only if there exists a biprincipal bibundle between them. Using a Hilsum-Skandalis tensor product, we further define a composition of diffeological bibundles, and …
We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topologica…
The purpose of this note is to define sheaves for diffeological spaces and give a construction of their Čech cohomology. As an application, we prove that the first degree Čech cohomology classes for the sheaf of smooth functions to an abelian diffeological group $G$ classify the diffeological principal $G$-bundles.
Diffeological Principal Bundles and Principal Infinity Bundles
本文研究微分学空间的∞-层上同调，对比现有Čech上同调概念，并证明微分学主G-丛与主∞-丛范畴神经弱同伦等价，沟通微分学丛理论与高阶拓扑斯理论。
Differential forms on diffeological spaces and diffeological gluing, I
本文研究微分拓扑空间沿光滑映射粘合时微分形式的性态，明确了微分形式的两种视角，重点刻画了全体微分形式构成的向量空间在粘合操作下的变化规律。
In this paper, we study a new operation named pushforward on diffeological vector pseudo-bundles, which is left adjoint to the pullback. We show how to pushforward projective diffeological vector pseudo-bundles to get projective diffeological vector spaces, producing many concrete new examples, together with application to smooth splittings of some projective diffeological vector spaces related to geometry. This brings new objects to diffeology from classical vector bundle theory.
We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart Λ^1(X) of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection on the exterior algebra pseudo-bundle \bigwedge(Λ^1(X)) with the standardly defined Clifford action by Λ^1(X); the latter is therefore assumed to admit a pseudo-metric for which there exists a Levi-Civita connection. Under these assumptions, the definition is fully analogous the…
We develop the theory of smooth principal bundles for a smooth group $G$, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define $D$-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a $D$-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor's spaces $EG$ and $BG$, show that $EG \to BG$ is a $D$-numerable principal bundle, and prove that it classifies all $D$-numerable principal bundles over …
Motivated by problems in which data are given over covering generating families, we suggest a new cohomology theory for diffeological spaces, called diffeological Čech cohomology, which is an exact $ \partial $-functor of the section functor for sheaves on diffeological spaces. As applications, under the situations of a setup, i) the generalized Mayer-Vietoris sequence for a diffeological space is established; ii) a version of the de Rham theorem is obtained, which connects diffeological Čech cohomology to the de Rham cohomology. Moreover, we characterize the isomorphism classes of diffeologic…
We give a new construction of the holonomy groupoid of a regular foliation in terms of a partial connection on a diffeological principal bundle of germs of transverse parametrisations. We extend these ideas to construct a novel holonomy groupoid for any foliated bundle, which we prove sits at the top of a hierarchy of diffeological jet holonomy groupoids associated to the foliated bundle.
Diffeological spaces firstly introduced by J.M. Souriau in the 1980s are a natural generalization of smooth manifolds. However, optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. This paper is devoted to an optimization technique o…
On the geometry of profinite diffeological spaces
本文研究profinite微分几何空间的几何性质，涵盖切余切空间、微分形式、度量、联络、算子及上同调关系，统一了多种构造。
A Diffeological Construction of Singer's Universal Connection
本文通过微分拓扑理论构建了Singer的通用连接，并将其推广到微分拓扑环境，展示了微分拓扑主丛与连接通过holonomy表示的重建，以及holonomy表示共轭的丛-连接对等价性。
We explore several notions of $k$-form at a point in a diffeological space, construct bundles of such $k$-forms, and compare sections of these bundles to differential forms. As they are defined locally, our $k$-forms can contain more information than the values of differential forms contain, and we illustrate this with many examples. To organize our work, we develop the basic theory of diffeological vector pseudo-bundles, including a detailed understanding of their limits and colimits, as well as a variety of fibrewise operations such as products, direct sums, tensor products, exterior powers …
We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural Cartan calculus, consisting of vector fields and differential forms, together with the Lie bracket, de Rham differential, inner derivative, and Lie derivative, satisfying the usual graded commutation relations. Elastic spaces are closed under arbitrary coproducts, finite products, and retracts. Examples include manifolds with corners and cusps, diffeological grou…
Lie algebras of quotient groups
研究商群$G/H$微分流到李代数的条件，并将经典不可积分巴纳赫-李代数整合到微分流群中。
Diffeological Tangent Spaces and Distributional Linearization for Lifted Euler--Reynolds Limits
该研究为不可压缩欧拉方程的欧拉-雷诺次解构建微分拓扑框架，通过提升型公式构造极限空间，建立一阶理论，引入有限混合模型实现切方向并得到相计数界与极小性结果。
Diffeological Spaces with a Non-Smooth Derivation
该研究展示了在某些微分流形上存在非光滑导出，表明切空间定义需超越单纯光滑导出，揭示光滑性无法仅由莱布尼茨法则恢复。
A diffeological perspective on non-Abelian Hodge theory
在紧Kähler流形上构造微分学模叠参数化Higgs丛和平坦丛，并建立子叠间的等价，该等价延拓了粗模空间同胚与范畴等价。
This manuscript attempts to present a way in which the classical construction of the Dirac operator can be carried over to the setting of diffeology. A more specific aim is to describe a procedure for gluing together two usual Dirac operators and to explain in what sense the result is again a Dirac operator. Since versions of cut-and-paste (surgery) operations have already appeared in the context of Atiyah-Singer theory, we specify that our gluing procedure is designed to lead to spaces that are not smooth manifolds in any ordinary sense, and since much attention has been paid in recent years …
We review the basic definitions and properties concerning smooth structures, convenient spaces, diffeological spaces and tangent structures. The relation betwen them is described. A tangent structure is constructed for each pre-convenient space. This one is proved to be convenient iff the space and the tangent fibres coincide. Some basic comments on the relation with manifolds are given.
Submersions, immersions, and étale maps in diffeology
本文通过非线性方法将淹没、浸入和平展映射适应于微分几何，引入微分几何平展流形以统一处理流形与无理环面，证明相关定理并简化内部切空间与维数的计算。
The Zariski Cotangent Space at the Origin of the Pencil Diffeological Space
本文研究仿射空间束在原点处的扎里斯基余切空间，证明其由过原点直线的导数决定，还计算了相关切空间，发现奇异点局部光滑结构可产生新拓扑概念。
Tiling spaces are covering spaces over irrational tori
本研究在微分拓扑框架下探究平铺空间，证明其基本微分拓扑性质，分析其在无理环面上的两种纤维丛结构，并借助无理环面的微分拓扑分类实现一维平铺空间的微分拓扑等价性继承。
In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal $q-$series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but t…
In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. With…
On spherical Milnor Classifying Spaces I: differential geometry
本文基于米尔诺构造，发展了差分几何中广义米尔诺分类空间的几何框架，引入了球面和投影模型，并研究了其切结构及边界行为，定义了微分形式和克利福德结构。
Spherical Milnor Spaces II: Projective Quotients and Higher Topological Structures
本文引入基于重心坐标二次规范化的球面米尔诺分类构造，用于微分流形群，生成具有群G和Z_2交换作用的合同微分流形，进而得到商空间层次结构，研究其拓扑和几何性质，包括射影模型和双商空间，用于研究带Z_2扭的主丛及低度上同调阻挠类。
Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle
本文否定微分拓扑空间上经光滑奇异复形转移模型结构的猜想，转而在光滑集层范畴建立适当组合、笛卡尔且与单纯集 Quillen 等价的模型结构，并推广光滑 Oka 原理以分类微分几何对象。
In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a Riemannian manifold $M$ which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to $T(M)$ a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on $T(M)$. We found conditions under which $T(M)$ is almost Kählerian, locally conformal Kählerian or Kählerian or when $T(M)$ has constant sectional curvature or constant scalar curvature. Then we will restric…
The purpose of the present work is to study the complete and horizontal lifts of the metallic structure on tangent bundles with respect to almost product structure. We also establish fundamental formulae related to integrability and horizontal lifts of metallic structures on tangent bundles. Moreover, the study revealed the behavior on cross-section of metallic structure in M to tangent bundle TM.
We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group $GL(\infty,\mathbb{R}) =\underrightarrow{\lim}GL(\mathbb{R}^{n})$.
In this article, we introduce some metallic structures on the tangent bundle of a P-Sasakian manifold by complete lift, horizontal lift and vertical lift of a P-Sasakian structure $(ϕ, η,ξ)$ on tangent bundle. Then we investigate the integrability and parallelity of these metallic structures.
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.
We describe conditions under which a spacetime connection and a scaled Lorentzian metric define natural symplectic and Poisson structures on the tangent bundle of the Einstein spacetime.
We classify locally the contact metric (k,mu)-spaces whose Boeckx invariant is $\le -1$ as tangent hyperquadric bundles of Lorentzian space forms.
The aim of this paper is to give some characterizations for N-Legendre and N-slant curves in the unit tangent bundles of surfaces endowed with natural diagonal lifted structures.
For a Riemannian manifold $M$, we determine some curvature properties of a tangent bundle equipped with the rescaled metric.The main aim of this paper is to give explicit formulae for the rescaled metric on $TM$, and investigate the geodesics on the tangent bundle with respect to the rescaled Sasaki metric.
Adopting the global approach to tangent bundles of order two established in[1], we develop this approach to find new results. We also generalize various results of [3], [4] and [6] to the geometry of tangent bundles of order two.
On the triviality of the generalized tangent bundle
研究了流形切包与广义切包的平凡性关系，证明平行可积流形的广义切包平凡，并通过莫比乌斯带、球面和射影空间例证逆向不成立，最后将广义切包平凡性与广义几何结构关联。
An isometric immersion of a Riemannian manifold M into a Riemannian manifold N gives rise in a natural way to the immersion of the tangent bundle TM into the tangent bundle TN with a non-degenerate g- natural metric G.
We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional curvature and any chosen constant r.
The main purpose of the paper is to investigate Killing vector field on the tangent bundle T(M_{n}) of the Riemannian manifold with respect to the Levi-Civita connection of the metric II+III .
In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent tangent bundles on 22-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal E$ be a vector bundle of rank $n$ over $M^n$, and $φ:TM^n\to \mathcal E$ an oriented vector bundle homomorphism. In this paper, we sh…
We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.
Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. …
On tangent sphere bundles with contact pseudo-metric structures
本文在切球丛上构造切触伪度量结构，并证明其成为(κ, μ)-切触伪度量流形或K-切触流形分别等价于基流形具有常截面曲率或常曲率ε。
In this paper we present a method by which is obtained a sequence of $k$-semisprays and two sequences of nonlinear connections on the $k$-tangent bundle $T^kM$, starting from a given one. Interesting particular cases appear for Lagrange and Finsler spaces of order $k$.
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