Abstract
1. Let X, Y be smooth finite dimensional manifolds, let C∞(X,Y) be the set of smooth mappings from X to Y; for any non negative integer n let Jn(X,Y) denote the fibre bundle of n-jets of smooth maps from X to Y, equipped with the canonical manifold structure which makes jnf : X → Jn(X,Y) into a smooth section for each f ∈c∞(X,Y) , where jnf(x) is the n-jet of f at xϵ X.
Usually C∞(X,Y) is equipped with the so called Whitney-C∞-topology: a basis of open sets is given by all sets of the form M(U) = {f ϵC∞(X,Y) : jnf (X)⊆ U} , where U is any open set in Jn(X,Y) and nϵN. See [ 3] and [6] for accounts of this topology. We may describe it intuitively by the following words: if you go to infinity on X you may control better and better partial derivatives up to a fixed order.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C. BESSAGA, A. PEŁCYNSKY: Slected topics in infinite dimensional topology, Polish scientific Publishers 1975.
H. CARTAN: Séminaire E.N.S. 1961/62: Topologie differen-tielle, éxposé by MORLET.
M. GOLUBITSKY, V. GUILLEMIN: Stable mappings and their singularities, Graduate texts in mathematics 14, Springer.
H.H. KELLER: Differential calculus in locally convex spaces, Springer lecture Notes 417 (1974).
J.A. LESLIE: On a differentiable structure for the group of diffeomorphisms, Topology 6 (1967), 263–271.
J. MATHER: Stability of C -mappings II: infinitesimal stability implies stability. Ann. Math. 89 (1969), 254–291.
P. MICHOR: Manifolds of smooth maps, to appear in Cahiers de topologie et géometrié differentielle. P. Michor, Mathematisches Institut der Universität, Strudlhofgasse 4, A-1090 Wien, Austria.
Editor information
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Michor, P. (2010). Manifolds of differentiable maps. In: Villani, V. (eds) Differential Topology. C.I.M.E. Summer Schools, vol 73. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11102-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-11102-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11101-3
Online ISBN: 978-3-642-11102-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)