Induced differential forms on manifolds of functions. (Q2897390)

Given finite dimensional manifolds \(S\) and \(M\), the set \(\mathcal {F}(S,M)\) of smooth functions \(S \to M\) is a Fréchet manifold. A relation between differential forms on \(S\), \(M\) and on \(\mathcal {F}(S,M)\) is studied. Specifically, using the projection map \(\text{pr}\: S \times \mathcal {F}(S,M) \to S\) and the evaluation map \(\text{ev}\: S \times \mathcal {F}(S,M) \to M\) and corresponding pullbacks, one has the map \(\Omega ^p(M) \times \Omega ^q(S) \to \Omega ^{p+q}(S \times \mathcal {F}(S,M))\) defined by \((\omega ,\alpha ) \mapsto \text{ev}^{*}\omega \wedge \text{pr}^{*}\alpha \) for \(\omega \in \Omega ^p(M)\) and \(\alpha \in \Omega ^q(S)\). Assuming \(S\) is compact oriented of the dimension \(k\) and integrating over the fiber \(S\) yields the hat pairing \(\Omega ^p(M) \times \Omega ^q(S)\to \Omega ^{p+q-k}(\mathcal {F}(S,M))\) defined as \(\widehat {\omega \cdot \alpha } \mapsto \int _S\text{ev}^{*}\omega \wedge \text{pr}^{*}\alpha \).NEWLINENEWLINENEWLINEThe aim of the article is to develop a hat calculus. In particular, it is shown the hat pairing descends to the pairing \(H^p(M) \times H^q(S) \to H^{p+q-k}(\mathcal {F}(S,M))\) on cohomologies and that the hat pairing is compatible with \(\text{Diff}(S)\)-actions on \(M\) and on \(\mathcal {F}(S,M)\) (and similarly for the \(\text{Diff}(M)\)-actions). Special cases \(\alpha =1\) (the identity) and \(\alpha \in \Omega ^k(S)\) (a volume form) define linear maps termed the hat map \(\Omega ^p(M) \to \Omega ^{p-k}(\mathcal {F}(S,M))\) and the bar map \(\Omega ^p(M) \to \Omega ^p(\mathcal {F}(S,M))\), respectively. Considering the natural map \(\text{Emb}(S,M) \to \text{Gr}_k(M)\) from the subset of embeddings \(\text{Emb}(S,M) \subseteq \mathcal {F}(S,M)\) to the non-linear Grassmannian \(\text{Gr}_k(M)\), the hat map recovers the tilda calculus on \(\text{Gr}_k(M)\) (see [\textit{S. Haller} and \textit{C. Vizman}, Math. Ann. 329, No. 4, 771--785 (2004; Zbl 1071.58005)]NEWLINENEWLINEFinally, symplectic structures and Hamiltonian actions are discussed in two cases. Assuming \(k=m-2\) where \(M\) is a closed manifold of the dimension \(m\) with a volume form \(\nu \in \Omega ^m(M)\), the hat map provides the symplectic structure \(\hat {\nu }\) on \(\text{Gr}_{m-2}(M)\). The group of exact volume preserving diffeomorphisms \(\text{Diff}_{\text{ex}}(M,\nu )\) acts in a Hamiltonian way on \((\text{Gr}_{m-2}(M),\hat {\nu })\). Further, assuming \((M,\omega )\) is a symplectic connected manifold and \(\alpha \in \Omega ^k(S)\) is a fixed volume form such that \(\int _S \alpha =1\), the bar map provides the symplectic form \(\bar {\omega }\) on \(\mathcal {F}(S,M)\). It is discussed how a Hamiltonian action on \(M\) gives rise to a Hamiltonian action on \(\mathcal {F}(S,M)\).