E Binz, H Fischer
1991 Note di Matematica   unpublished
The present contribution to this volume is concemed with certain problems in non-linear functional analysis which are motivated by classica1 physics, specifically by elasticity theory: we are given a «body», i.e. a compact smooth manifold M' which moves and may be deformed in some Rn (equipped with a fixed inner product); we assume that the motion and deformation are such that the diffeomorphism type of M' does not change. Hence, M' is the image under a smooth embedding of some compact smooth
more » ... nifold M (possibly with boundary 8M) and the appropriate configuration space for the problem is the set E(M, R ") of smooth embeddings M + R n ; this set is a smooth Frechet manifold when endowed with its natura1 C"-topo1ogy. The deformable medium is to be characterized by a «smooth one-form» on E(M, R ") , i.e. by a smooth real-valued function F which to each configuration J E E(M, R ") and distortion L E CJ"(M, R ") assigns a number F(J)(L) , depending linearly on L, which is interpreted as the work caused by L at J, cf. section 4. An approach to elasticity along these lines is described e.g. in [E,S] and [Bi 41; cf. also section 6 for more details where we also relate our treatment to the usual one such as given in [L,L]. If the deformations mentioned above are subject to smooth constraints or if the motion no longer takes piace in R n, we will still assume that the ambient space is a smooth Riemannian manifold N and this forces us to introduce as a configuration space the manifold E(M, N) of smcoth embeddings M-f N. Since the tangent bundle TE(M, N) no longer is trivial, in general, the treatment of one-forms on E(M, N) becomes somewhat more complicated. In order to obtain «integrai representations» of certain one-forms, we assume that both M and N are oriented. With this assumption, sections 2 and 3 introduce the basic geometrie ingredients needed for integra1 rcpresentations of those one-forms which at each ,7 E E(M, N) only depend on the one-jets of the vector fields L «along J ». We introduce the metrics @I and p a on E(M, N) and E(i3M, N) , respectively, which are continuous, symmetric and positive-definite bilinear forms on the respective tangent spaces. Both p and p s are invariant under the group Diff'M of orientation preserving diffcomorphisms of M and any group g of orientation preserving isometries of N. Section 3 furthermore introduces the bundle p E(M, TN) of «smooth TN-valued one-forms on M » which cover embeddings M t N, fibred over E(M, N) by the Fréchet spaces