Tangent space and differential on a Banach manifold

Let $k\ge 1$, let $M$ be a $C^k$ Banach manifold modelled on the real Banach space $E$ (Countable base Banach manifold and smooth map) and let $p\in M$. Consider the set of pairs $(\phi ,v)$ in which $\phi$ is a chart of $M$ whose domain contains $p$ and $v\in E$, and declare

$(\phi ,v)\sim (\psi ,w):⟺w=D(\psi \circ {\phi }^{-1})(\phi (p))v,$

the derivative being that of the transition map, a $C^1$ map between open subsets of $E$ (Fréchet derivative between Banach spaces). The tangent space to $M$ at $p$ is the quotient set

$T_{p}M:=\{(\phi ,v):p\in \mathrm{dom}\phi \}/\sim ,$

and the class of $(\phi ,v)$ is written $[\phi ,v]$. For a chart $\phi$ at $p$ the assignment $[\phi ,v]\mapsto v$ identifies $T_{p}M$ with $E$; the resulting real vector space structure is

$\lambda [\phi ,v]+\mu [\phi ,w]:=[\phi ,\lambda v+\mu w](\lambda ,\mu \in \mathbb{R}),$

and the differential of a $C^1$ map $f:M\to N$ between Banach manifolds (models $E$ and $F$) at $p$ is

$Df(p)([\phi ,v]):=[\psi ,D(\psi \circ f\circ {\phi }^{-1})(\phi (p)\left)v\right]\in T_{f(p)}N,$

where $\phi$ is any chart of $M$ at $p$ and $\psi$ any chart of $N$ at $f(p)$ for which the representative $\psi \circ f\circ {\phi }^{-1}$ is defined near $\phi (p)$. The well-definedness of the relation, of the vector space operations and of the differential, together with the identities below, is proved on this page as Banach manifold differentials are chart independent.

• Tangent vectors are velocities of curves. If $\phi$ is a chart at $p$ and $v\in E$, then the curve $\gamma (t):={\phi }^{-1}(\phi (p)+tv)$, defined for small real $t$, lies in $M$ and satisfies $\phi \circ \gamma (t)=\phi (p)+tv$, so its coordinate velocity at $0$ is $v$; the class $[\phi ,v]$ is exactly that velocity. Conversely every velocity of a curve through $p$ arises in this way. This is the reading used in the counterexample on the companion page, where a curve in a closed subspace produces a tangent vector of the subspace.

• The differential is linear on tangent spaces. This is not part of the definition but follows from the chain rule: in a fixed chart at $p$ and a fixed chart at $f(p)$ the map $v\mapsto D(\psi \circ f\circ {\phi }^{-1})(\phi (p))v$ is bounded linear, and the chart identifications are linear. The functoriality statements $D(\mathrm{i}\mathrm{d})=\mathrm{i}\mathrm{d}$ and $D(g\circ f)=Dg\circ Df$ are proved with the same computation.

• The vector space structure does not depend on the chart. A change $\phi \to \psi$ multiplies coordinate vectors by the transition derivative $D(\psi \circ {\phi }^{-1})(\phi (p))$, which is a bounded linear isomorphism of $E$ with inverse $D(\phi \circ {\psi }^{-1})(\psi (p))$; linearity of this change is exactly what makes the displayed operations independent of the chart chosen. The invertibility follows from the chain rule: the two transition maps are mutually inverse $C^1$ maps, so their composites are the identity on open sets and differentiating those identities exhibits each derivative as the inverse of the other. Both facts are recorded in the lemma below.

• For an admissible open model the tangent space is the model space. If the norm topology of $E$ is second countable, $W\subseteq E$ is open, and $p\in W$, then $W$ is a Banach manifold under the convention of Countable base Banach manifold and smooth map. The single chart $({\mathrm{i}\mathrm{d}}_W,W)$ makes $T_{p}W$ the set of classes $[{\mathrm{i}\mathrm{d}}_W,v]$, which is canonically identified with $E$; under this identification $Df(p)$ of a map $f:W\to E$ is the Fréchet derivative of the coordinate representative, which here is $f$ itself. All computations on this page are performed through this identification.