Kefeng Liu, Xiaonan Ma
In this note we verify certain statement about the operator $Q\_K$ constructed by Donaldson in [3] by using the full asymptotic expansion of Bergman kernel obtained in [2] and [4].
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We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.
This thesis introduces the notion of "relative gerbes" for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are classified by the relative integral cohomology in degree three. Furthermore, by using the concept of relative gerbes, the pre-quantization of Lie group-valued moment maps is developed, and its equivalence with the pre-quantization of infinite-dimensional Hamiltonian loop group spaces is established.
In this article, we treat G_2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G_2-structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G_2-structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.
In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between $T$-dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.
Let $M$ be a projective toric manifold. We prove two results concerning respectively Kaehler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C^n.
We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].
In this note we prove some results in flat and differential $K$-theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential $K$-theory and Freed-Lott differential $K$-theory.
Using the navigation data (h,W) of a Kropina space, we characterize weakly-Berwald Kropina spaces and Berwald Kropina spaces by means of the Killing vector field W and the parallel vector field W, respectively. Moreover, the local 1-parameter group of Finslerian local isometries of the Kropina space coincides with the local 1-parameter group Riemannian of local isometries on (M, h).
A connection between differential geometry and soliton equations is discussed
In this paper we give an axiomatization of differential geometry comparable to model categories for homotopy theory. Weil functors play a predominant role.
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be replaced by functors on the category of Weil algebras. Our model theory is geometrically natural and conceptually motivated, while the model theory of synthetic differential geometry is highly artificial and exquisitely technical.
Given a special Kahler manifold M, we give a new, direct proof of the relationship between the quaternionic structure on its cotangent bundle and the variation of Hodge structures on the complexification of TM.
Some remarks on fibrations in complex geometry
本文探讨了复解析背景下全纯纤维化的若干性质，为复几何研究提供了补充注记。
We refurbish our axiomatics of differential geometry introduced in [Mathematics for Applications,, 1 (2012), 171-182]. Then the notion of Euclideaness can naturally be formulated. The principal objective in this paper is to present an adaptation of our theory of differential forms developed in [International Journal of Pure and Applied Mathematics, 64 (2010), 85-102] to our present axiomatic framework.
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned with differential-geometric developments within the above axiomatic scheme, this paper is devoted to vector fields. The principal result is that the totality of vector fields on a microlinear and Weil exponential object forms a Lie algebra.
We give simple upper and lower bounds for the order of a Klein geometry
In [21], the authors initiated the study of quasi generalized CR (QGCR)-null submanifolds. In this paper, attention is drawn to some distributions on ascreen QGCR-null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifold.
Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.
A remark on the reach and upper bounds on some extrinsic geometry invariants of submanifolds
本文利用第二变分公式研究黎曼流形中紧子流形的可达距离，建立其与外在几何不变量之间的关系，并将欧氏空间中的相关定理推广到一般黎曼流形。
In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.
Given a complete and (locally) cartesian closed category U, it is shown that the category of functors from the category of Weil algebras to the category U is (locally, resp.) cartesian closed. The corresponding axiomatization for differential geometry based upon Weil functors is then given.
In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just as grad (ient), div (ergence) and rot (ation) are uniquely determined geometrically or physically in classical vector calculus. This infinitesimal characterization of exterior differentiation has been completely missing in orthodox differential geometry.
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontriv…
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize distribution spaces in our study of wave and heat equations.
As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.
In this paper, we establish equiform differential geometry of space and timelike curves in 4-dimensional Minkowski space. We obtain some conditions for these curves. Also, general helices with respect to their equiform curvatures are characterized.
Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.
This is the first part of the lecture notes that grew out of the special course given during the 2021-2022 academic year. In these lecture notes we present an approach to the fundamental structures of differential geometry that uses the vernacular of sheaves, differential operators and horizontal subbundles.
The physics of classical particles in a Lorentz-breaking spacetime has numerous features resembling the properties of Finsler geometry. In particular, the Lagrange function plays a role similar to that of a Finsler structure function. A summary is presented of recent results, including new calculable Finsler structures based on Lagrange functions appearing in the Lorentz-violation framework known as the Standard-Model Extension.
For a more general notion of Cartan connection we define characteristic classes, we investigate their relation to usual characteristic classes.
In this paper we investigate the balanced condition (in the sense of Donaldson) and the existence of an Englis expansion for the LeBrun's metrics on $C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balanced unless it is the flat metric. The second one shows that an Englis expansion of the Rawnsley's function associated to a LeBrun's metric always exists, while the coefficient $a_3$ of the expansion vanishes if and only if the LeBrun's metric is indeed the flat one.
This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.
The principal objective in this paer is to study the relationship between the old kingdom of differential geometry (the category of smooth manifolds) and its new kingdom (the category of functors on the category of Weil algebras to some smooth category). It is shown that the canonical embedding of the old kingdom into the new kingdom preserves Weil functors.
The fourth paper of our series of papers entitled "Differential Geometry of Microlinear Frolicher Spaces is concerned with jet bundles. We present three distinct approaches together with transmogrifications of the first into the second and of the second to the third. The affine bundle theorem and the equivalence of the three approaches with coordinates are relegated to a subsequent paper.
Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. We define discret…
This paper has been withdrawn by the authors due to the fact that the webs considered in the paper are ``Veronese-like webs'' which are different from Veronese webs.
In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to affine congruence.
Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.
Survey talk on certain aspects of the subject, stressing the neighbor relation as a basic notion in differential geometry.
This book is a textbook for the basic course of differential geometry. It is recommended as an introductory material for this subject.
We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$. The framework is that of the tangent sphere bundle of $M$. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler-Lagrange differential systems.
In this work, differential geometry of the Z$_3$-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geometry. Moreover, we also present recent differential geometric results on maps and immersions which are constructed in ways similar to Segre embedding.
As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity. It is also shown that the Lie derivation preserves the Frolicher-Nijenhuis bracket. Our definitions and discussions are highly geometric, while Frolicher and Nijenhuis' original definitions and discussions were highly algebraic.
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of even basic concepts such as discrete curvature κ, discrete torsion τ, or discrete Frenet frame.
Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z$_2$-graded Hopf algebra structure is obtained. Its Z$_2$-graded dual Hopf algebra is also given.
This is the lecture 4 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\mathbb{R}^{2}$. Here we use the Cartan method t…
This is the lecture 3 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\mathbb{R}^{2}$. Here we use the Cartan method t…
This is the lecture 1 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\mathbb{R}^{2}$. Here we use the Cartan method t…
This is the lecture 2 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\mathbb{R}^{2}$. Here we use the Cartan method t…
Remarks on Lipschitz geometry on globally conic singular manifolds
该研究探讨带锥形奇异性的流形的度量性质，建立度量锥形与渐近锥形行为的关联，证明满足奇点锥形、端渐近锥形条件的环境奇异流形中的奇异子流形是Lipschitz正规嵌入的，解决了此前提出的相关问题。
The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have emancipated microlinearity from within well-adapted models to Frolicher spaces. Therein we have shown that Frolicher spaces which are microlinear as well as Weil exponentiable form a cartesian closed category. To make sure that such Frolicher spaces are the central object of infinite-dimensional differential geometry, we develop the the…
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on quadratic differentials that lead to analogue of half-translation structures on spherical CR manifolds. Finally, we work on known examples of quasiconformal maps in the Heisenberg group with extremal properties and explicit how quadratic differentials are involved in those. In add…
We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on some aspects of contact geometry.
We construct a right-invariant differential calculus on the quantum supergroup GL$_h(1| 1)$ and obtain the $h$-deformed superalgebra of GL$_h(1| 1)$.
We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this approach with explicit examples. In particular, we describe a short program to determine whether a given linear exterior differential system is involutive.
These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting from the Cartan connection, recognition of symmetric 2nd order ODEs by their differential invariants and the constructive aspects of the local equivalence problem.
What is differential geometry: curves and surfaces
本书通过曲线和曲面介绍微分几何，提供基础技术与可视化方法，帮助读者理解该领域的核心概念与研究方向。
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