Alexey Basalaev, Claus Hertling
F-manifolds are complex manifolds with a multiplication with unit on the holomorphic tangent bundle with a certain integrability condition. Here the local classification of 3-dimensional F-manifolds with or without Euler fields is pursued.
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We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.
Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces.
We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.
This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible flat structures on them furnish a useful weakening of Dubrovin's Frobenius structure which naturally arises in the quantum $K$--theory, theory of extended moduli spaces, and unfolding spaces of singularities.
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the firs…
Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $π_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.
On compact Kähler manifold with strongly pseudo-effective tangent bundle
本文研究紧致凯勒流形的强伪有效切丛，证明大切丛仅存在于射影空间，并刻画 nef 向量丛。
In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general non-minimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.
We study minimal rational curves on a complex manifold that are tangent to a distribution. In this setting, the variety of minimal rational tangents (VMRT) has to be isotropic with respect to the Levi tensor of the distribution. Our main result is a converse of this: any smooth projective variety isotropic with respect to a vector-valued anti-symmetric form can be realized as VMRT of minimal rational curves tangent to a distribution on a complex manifold. The complex manifold is constructed as a partial equivariant compactification of a metabelian group, which is a result of independent intere…
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 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.
A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.
We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.
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.
On compact Kähler manifolds with pseudo-effective tangent bundle
本文证明了具有伪有效或奇异正曲率切丛的紧Kähler流形存在有理连通纤维化结构，将光滑射影簇的结构定理推广至紧Kähler流形。
We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle still preserve the property to have all geodesic curvatures constant.
We give an intrinsic (coordinate-free) construction of the tangent groupoid of a filtered manifold.
It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension 4 with non-Euclidean volume growth who has at least two distinct tangent cones at infinity and one of them has a smooth cross section.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^*X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is a Fano manifold such that $P(T^*X)$ has nef tangent bundle, then $X\cong P^n$.
In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold $M\times\mathbb{R}$, where $M$ is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the poi…
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples, and find the cohomological obstractions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, wh…
Geodesics and $F$-geodesics on tangent bundle with $φ$-Sasaki metric over para-Kähler-Norden manifold
本文研究仿凯勒-诺登流形上配备φ-Sasaki度量的切丛及φ-单位切丛中的测地线与F-测地线问题。
The geometry of the tangent and sphere bundles over statistical manifolds
本文利用底流形上的统计结构定义了切丛的黎曼结构，推导其曲率并证明刚性结果，进而开创性地研究了统计流形上的球丛几何。
In this paper, we introduce the notion of a super tangent bundle of a manifold, and extend the basic notions of differential geometry such as differential forms, exterior derivation, connection, metric and divergence on manifolds that equipped with a super tangent bundle.
On natural symmetries on slit tangent bundles of Finsler manifolds
本文在Finsler流形的slit切丛上引入$F$-自然度量，并刻画其共形、位似和Killing向量场，推广了黎曼流形上的$g$-自然度量理论。
Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We…
The present paper unifies some aspects concerning the vertical Liouville distributions on the tangent (cotangent) bundle of a Finsler (Cartan) space in the context of generalized geometry. More exactly, we consider the big-tangent manifold $\mathcal{T}M$ associated to a Finsler space $(M,F)$ and of its $\mathcal{L}$-dual which is a Cartan space $(M,K)$ and we define three Liouville distributions on $\mathcal{T}M$ which are integrable. We also find geometric properties of both leaves of Liouville distribution and the vertical distribution in our context.
We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $η$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $η$-Einstein manifold if and only if base manifold is the space of constant sectional curvature 1 or 2.
Geometry Via Sprays on Frechet Manifolds
本文利用喷射概念在 Fréchet 流形的切丛和二阶切丛上构造并刻画线性对称联络，证明其与喷射之间存在双射对应。
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。
The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. Here we extend this result to $M$ modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems…
We define a class of metrics that extend the Sasaki metric of a tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback of the big tangent bundle of a manifold to the tangent manifold. We obtain the expression of the transferred metric and we 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 Kaehler structures.
Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the action is locally free on the cross section. This generalizes the metric cone theorem of Cheeger-Colding to the Kähler case. We also discuss some applications to complete Kähler manifolds with nonnegative bisectional curvature.
We study manifolds arising as spaces of sections of complex manifolds fibering over the projective line with normal bundle of each section isomorphic to several copies of O(k). Such manifolds provide a natural setting for certain integrable systems, "monopoles", which comprise the Bogomolny hierarchy and the self-dual hierarchy.
As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid $\cG_{H} M$ encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of a refined …
A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold, is a straightforward application. A historical account on the subject is provided in order to enrich the mathematical panorama. Old characterizations of smooth manifold (by tangent cones), due to Valiron (1926, 1927) and Severi (1929, 1934) are recovered; modern characterization…
On the slit tangent manifold $TM^0$ of a Finsler space $(M,F)$ there are given some natural foliations as vertical foliation and some other fundamental foliations produced by the vertical and horizontal Liouville vector fields, see [A. Bejancu, H. R. Farran, Finsler Geometry and Natural Foliations on the Tangent Bundle, Rep. Math. Physics 58, No. 1 (2006), 131-146]. In this paper we consider a $(n,2n-1)$-codimensional subfoliation $(\mathcal{F}_V,\mathcal{F}_Γ)$ on $TM^0$ given by vertical foliation $\mathcal{F}_V$ and the line foliation spanned by vertical Liouville vector field $Γ$ and we gi…
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. …
Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. At the end we introduce the notion of the holonomy algebra of a Finsler ma…
We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $ω$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if a…
The tangent bundle of an almost Norden manifold and the complete lift of the Norden metric is considered as a 4n-manifold. It is equipped with an almost hypercomplex Hermitian-Norden structure. It is characterized geometrically. The case when the base manifold is an h-sphere is considered.
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。
We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.
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.
Let $G$ be a connected complex Lie group and $Γ\subset G$ a cocompact lattice. Let $H$ be a complex Lie group. We prove that a holomorphic principal $H$-bundle $E_H$ over $G/Γ$ admits a holomorphic connection if and only if $E_H$ is invariant. If $G$ is simply connected, we show that a holomorphic principal $H$-bundle $E_H$ over $G/Γ$ admits a flat holomorphic connection if and only if $E_H$ is homogeneous.
We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kählerian Ricci-flat manifolds in four real dimensions.
Coarse pseudo-differential calculus and index theory on manifolds with a tangent Lie structure
本研究针对完备黎曼流形零阶算子提出简化粗伪微分演算，将其拓展至具切李结构的广义流形，并证明h-椭圆算子的指标为流形K同调群元素的指标定理。
In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.
We show that the holonomy invariance of a function on the tangent bundle of a manifold, together with very mild regularity conditions on the function, is equivalent to the existence of local parallelisms compatible with the function in a natural way. Thus, in particular, we obtain a characterization of generalized Berwald manifolds. We also construct a simple example of a generalized Berwald manifold which is not Berwald.
On Deformation Spaces, Tangent Groupoids and Generalized Filtrations of Banach and Fredholm Manifolds
本文研究了Banach和Fredholm流形的广义滤波，并扩展了切群体和切丛的构造方法，提出了更灵活的理论框架。
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…
Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometricians. Here we define a Riemannian or pseudo-Riemannian lift metric on TM, which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on its tangent bundle with this metric and prove among the others that, every complete lift conformal vector field on TM is homothetic, and…
A Born Structure on the Tangent Bundle of a Hessian Manifold
本文研究Hessian流形切丛上的几何结构，探讨了仿射联络与黎曼度量构成的有序对定义Hessian结构的条件，和其诱导的殆Born结构可积条件之间的等价性，关联了两类几何结构的研究。
The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately. The equivalence of the existence of Killing vector field on M and TM is proved. Key words: Riemannian manifold, tangent bundle, g-natural metric, Killing vector field, non-degenerate metric.
We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by bilipschitz maps. Furthermore, $M$ is countably $N$--rectifiable, i.e., all of $M$ except for a null set can be covered by countably many such maps.
A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentiall…
We study the geometry of pseudo-Riemannian manifolds which are Jacobi--Tsankov, i.e. J(x)J(y)=J(y)J(x) for all tangent vectors x and y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. J(x)J(y)=0 for all tangent vectors x and y.
Geometry of tangent bundles of statistical manifolds equiped with Cheeger-Gromoll type metrics
本文研究统计流形的切丛在Cheeger-Gromoll型度量下的几何特性，计算Levi-Civita联络及其曲率，分析测地线行为，确定纤维全测地性条件，并探讨几何流的不可压缩性。
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