اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدManifolds of mappings for continuum mechanics
76
0
0.0
(
0
)
تأليف
Peter W. Michor
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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.
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 work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different meth
od, we generalize maximum principles of Omori and Yau to a viscosity version.
Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $text{Nil}$. We show that the diffeomorphism group $text{Diff}(X)$ deformation retracts to the isometry group $text{Isom}(X)$. Combining this with earlier work by m
any authors, this completes the determination the homotopy type of $text{Diff}(X)$ for any compact, orientable, prime 3-manifold $X$.
Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the ONeill t
ensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kahler-Einstein manifold or a Riemannian product of a Kahler-Einstein manifold with R (or with the circle S^1).
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
