Retained and omitted variation spaces

All Banach spaces below are real complete normed vector spaces. A $C^1$ Banach manifold is a Hausdorff space with charts to open subsets of real Banach spaces whose transition maps are $C^1$; a map is $C^1$ when its chart representatives have Fréchet derivatives that are continuous in operator norm. In charts, $D{F}_x$ is the bounded linear map for which $F(x+h)=F(x)+D{F}_{x}h+o(\mathrm{\parallel }h\mathrm{\parallel })$. Tangent vectors are model vectors identified by derivatives of chart transitions, and $T_x^{\ast }$ is the space of bounded real-linear functionals on $T_x$. A topological direct sum $T=V\oplus W$ means that addition $V\times W\to T$ is a bounded linear isomorphism.

Let $X$ and $X_r$ be specified smooth manifolds, possibly with boundary, and let the full and retained fields be sections of specified smooth bundles over them. Choose integers $k\ge 1$ sufficient for the chosen action and boundary data. Let $\mathcal{C}$ and ${\mathcal{C}}_r$ be specified $C^1$ Banach manifolds of admissible $C^k$ full and retained fields, respectively; their boundary conditions and gauge fixing, or their smooth gauge quotients, are part of these spaces. Let $S:\mathcal{C}\to \mathbb{R}$ and $I:{\mathcal{C}}_r\to \mathcal{C}$ be $C^1$, and fix $\varphi \in {\mathcal{C}}_r$ with $\psi =I(\varphi )$. The admissible full variations are exactly $T_{\psi }\mathcal{C}$, and the retained variations are $V_{\mathrm{r}\mathrm{e}\mathrm{t}}(\varphi ):=\mathrm{im}D{I}_{\varphi }\subseteq T_{\psi }\mathcal{C}\mathrm{.}$ Suppose a closed subspace $V_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}(\varphi )\subseteq T_{\psi }\mathcal{C}$ has been specified such that $T_{\psi }\mathcal{C}=V_{\mathrm{r}\mathrm{e}\mathrm{t}}(\varphi )\oplus V_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}(\varphi )$ as a topological direct sum. The reduced action is $S_r:=S\circ I$. Its derivative is the continuous covector $D{S}_r{\mathrm{\mid }}_{\varphi }=(D{I}_{\varphi }{)}^{\ast }(DS{\mathrm{\mid }}_{\psi }),$ where $(D{I}_{\varphi }{)}^{\ast }:T_{\psi }^{\ast }\mathcal{C}\to T_{\varphi }^{\ast }{\mathcal{C}}_r$ is the continuous-dual pullback. For every $w\in T_{\varphi }{\mathcal{C}}_r$, $D{S}_r{\mathrm{\mid }}_{\varphi }(w)=DS{\mathrm{\mid }}_{\psi }(D{I}_{\varphi }w)$, so $D{S}_r{\mathrm{\mid }}_{\varphi }=0$ if and only if $DS{\mathrm{\mid }}_{\psi }$ annihilates $V_{\mathrm{r}\mathrm{e}\mathrm{t}}(\varphi )$. The reduced covector and the restriction of $DS{\mathrm{\mid }}_{\psi }$ have different domains in general; they are not identified unless $D{I}_{\varphi }$ identifies $T_{\varphi }{\mathcal{C}}_r$ with its image. The omitted first variation is the restriction of $DS{\mathrm{\mid }}_{\psi }$ to $V_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}(\varphi )$. The stated tangent-space split, boundary conditions, and gauge treatment are part of the data; without a declared complement there is no defined omitted variation space.