We describe the notion of a \emph{weighting} along a submanifold $N\subset M$, and explore its differential-geometric implications. This includes a detailed discussion of weighted normal bundles, weighted deformation spaces, and weighted blow-ups. We give a description of weightings in terms of subbundles of higher tangent bundles, which leads us to notions of multiplicative weightings for Lie algebroids and Lie groupoids.
Axiomatic Differential Geometry II-2: Differential Forms
我们重新定义了[Mathematics for Applications，，1（2012），171-182]中介绍的微分几何公理化。于是，欧几里德性的概念就可以自然地表述出来了。本文的主要目的是提出一个适应我们的理论的微分形式发展[国际纯粹与应用数学杂志，64（2010），85-102]，我们目前的公理化框架。
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 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.
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.
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.
Differential Geometry of Microlinear Frolicher Spaces I
综合微分几何的中心对象是微线性空间。在我们之前的论文[Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-，International Journal of Pure and应用数学，60（2010），15-24]中，我们已经将微线性从适应良好的模型中解放到Frolicher空间。在这里，我们已经表明，Frolicher空间是微线性的，以及Weil指数形成一个carnival闭范畴。为了使这类Frolicher空间成为无穷维微分几何的中心对象，本文发展了这类空间上的向量场理论。中心结果是这样一个Frolicher空间上的所有向量场形成一个李代数。
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…
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.
We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form for the minimal Lagrangian surfaces. The local geometry, at least for main cases of interest, induces a natural frame whose structure equations arise from the affine Toda system for $\mathfrak a^{(…
This is the second paper in a series devoted to developing an arithmetic PDE analogue of Riemannian geometry. In Part 1 arithmetic PDE analogues of Levi-Civita and Chern connections were introduced and studied. In this paper arithmetic analogues of curvature and characteristic classes are developed.
