Эта публикация цитируется в
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Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds
M. Michelia,
P. W. Michorb,
D. Mumfordc a Université René Descartes
b University of Vienna
c Brown University
Аннотация:
Given a finite-dimensional manifold
$N$, the group
$\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphisms diffeomorphism of
$N$ which decrease suitably rapidly to the identity, acts on the manifold
$B(M,N)$ of submanifolds of
$N$ of diffeomorphism-type
$M$, where
$M$ is a compact manifold with
$\operatorname{dim} M<\operatorname{dim} N$. Given the right-invariant weak Riemannian metric on
$\operatorname{Diff}_{\mathcal S}(N)$ induced
by a quite general operator $L\colon \mathfrak X_{\mathcal S}(N)\to \Gamma(T^*N\otimes\operatorname{vol}(N))$,
we consider the induced weak Riemannian metric on
$B(M,N)$ and compute its geodesics and sectional curvature.
To do this, we derive a covariant formula for the curvature in finite and infinite dimensions, we show how
it makes O'Neill's formula very transparent, and we finally use it to compute the sectional curvature on
$B(M,N)$.
Bibliography: 15 titles.
Ключевые слова:
robust infinite-dimensional weak Riemannian manifolds, curvature in terms of the cometric, right-invariant Sobolev metrics on diffeomorphism groups, O'Neill's formula, manifold of submanifolds.
УДК:
514.83+
517.988.24
MSC: 58B20,
58D15,
37K65 Поступило в редакцию: 16.02.2012
Язык публикации: английский
DOI:
10.4213/im7966
Полный текст:
PDF файл (650 kB)
