اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدManifolds of mappings and shapes
130
0
0.0
(
0
)
تأليف
Peter W. Michor
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of manifolds of smooth mappings. Smooth manifolds of immersions, diffeomorphisms, and shapes, and weak Riemannian metrics on them are treated, culminating in the surprising fact, that geodesic distance can vanish completely for them.
قيم البحث
اقرأ أيضاً
This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{infty}(M,N)$ of all smooth
mappings from a finite dimensional Whitney manifold germ $M$ into a smooth manifold $N$. A Whitney manifold germ is a smooth (in the interior) manifold with a very general boundary, but still admitting a continuous Whitney extension operator. This notion is developed here for the needs of geometric continuum mechanics.
Given smooth manifolds $M_1,ldots, M_n$ (which may have a boundary or corners), a smooth manifold $N$ modeled on locally convex spaces and $alphain({mathbb N}_0cup{infty})^n$, we consider the set $C^alpha(M_1timescdotstimes M_n,N)$ of all mappings $f
colon M_1timescdotstimes M_nto N$ which are $C^alpha$ in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $leq alpha_j$ in the $j$th variable for $jin{1,ldots, n}$, in local charts. We show that $C^alpha(M_1timescdotstimes M_n,N)$ admits a canonical smooth manifold structure whenever each $M_j$ is compact and $N$ admits a local addition. The case of non-compact domains is also considered.
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at
infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.
Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincare inequality, the space $Mone$, defined by Haj{l}asz, is identified with a Hardy-Sobolev space defined
in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.
The projective shape of a configuration of k points or landmarks in RP(d) consists of the information that is invariant under projective transformations and hence is reconstructable from uncalibrated camera views. Mathematically, the space of project
ive shapes for these k landmarks can be described as the quotient space of k copies of RP(d) modulo the action of the projective linear group PGL(d). Using homogeneous coordinates, such configurations can be described as real k-times-(d+1)-dimensional matrices given up to left-multiplication of non-singular diagonal matrices, while the group PGL(d) acts as GL(d+1) from the right. The main purpose of this paper is to give a detailed examination of the topology of projective shape space, and, using matrix notation, it is shown how to derive subsets that are in a certain sense maximal, differentiable Hausdorff manifolds which can be provided with a Riemannian metric. A special subclass of the projective shapes consists of the Tyler regular shapes, for which geometrically motivated pre-shapes can be defined, thus allowing for the construction of a natural Riemannian metric.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
