Intrinsic characterization of manifold-valued generalized functions

Generalized functions valued in a smooth manifold

Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalized functions and indicate applications of the resulting theory in general relativity.

Lifting for manifold-valued maps of bounded variation

Let $\mathcal{N}$ be a smooth, compact, connected Riemannian manifold without boundary. Let $\mathcal{E}\to\mathcal{N}$ be the Riemannian universal covering of $\mathcal{N}$. For any bounded, smooth domain $Ω\subseteq\mathbb{R}^d$ and any $u\in\mathrm{BV}(Ω, \, \mathcal{N})$, we show that $u$ has a lifting $v\in\mathrm{BV}(Ω, \, \mathcal{E})$. Our result proves a conjecture by Bethuel and Chiron.

Topological singular set of vector-valued maps, I: Applications to manifold-constrained Sobolev and BV spaces

We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a closed submanifold $N$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $N$ is a sphere. The range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $N$-valued maps with respect to Sobolev norm…

Sheaves of nonlinear generalized functions and manifold-valued distributions

This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space $\mathcal{G}[X,Y]$ of Colombeau generalized functions defined on a manifold $X$ and taking values in a manifold $Y$. This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings $\mathcal{C}(X,Y)$ into $\mathcal{G}[X,Y]$ and study the sheaf properties of $\mathcal{G}[X,Y]$. Similar results are obtained for spaces of generalized vector bundle hom…

形式流形上的函数空间

Function spaces on formal manifolds

作为形式流形相关系列研究的一部分，本文在已有形式流形概念与基础框架的基础上，研究形式流形上的各类函数空间，实现光滑流形上向量值广义函数与分布的推广。

The index of Toeplitz operators on compact Lie groups and simply connected closed 3-manifolds

In this paper we use the notion of operator-valued symbol in order to compute the index of Toeplitz operators on compact Lie groups. Our approach combines the Connes index theorem and the infinite-dimensional operator-valued symbolic calculus of Ruzhansky-Turunen. We also give applications to the index of Toeplitz operators on simply connected closed $3$-manifolds $\mathbb{M}\simeq \mathbb{S}^3\simeq \textnormal{SU}(2) ,$ by using, as a fundamental tool, the Poincaré theorem (see Perelman [31,32,33,34])

可被光滑映射弱逼近的流形值映射的拓扑奇异集

Topological point singularities of manifold-valued maps and approximability by smooth maps

研究从(p+1)维欧氏区域到闭黎曼流形N的W^{1,p}映射，在π_1(N)在π_p(N)上作用平凡条件下，利用π_p(N)系数的平坦链刻画弱序列闭包中映射的点奇异，其消失刻画局部强可光滑逼近性。

Intrinsic weak derivatives and Sobolev spaces between manifolds

We define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various i…