nLab
local addition

Contents

Idea

At its simplest, a smooth manifold is a place where one can do calculus without worries. Let us consider the formula for a derivative:

f(x)=lim h0f(x+h)f(x)h f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

From the fractional part of this, it appears that to be able to discuss calculus, we need to be working in a space which allows addition and scalar multiplication. However, the fact that we are considering a limit means that we only need to be able to do such things in a neighbourhood of a given point. That is, we need the concept of a local addition.

Thus at its most general, a local addition at a point is simply the structure of a (piece of a) vector space near that point. This is, of course, the notion of a chart. At this point, one starts to think about notions of compatibility between different local additions. By concentrating on what it means for two charts to be compatible at a point, one soon arrives at the notion of an atlas.

Another direction that one could take is to ask for a system of local additions, one for each point, and ask that these fit together in some nice manner as one moves over the manifold. This leads to the notion of a local addition.

Definition

The broadest definition is the following.

Definition

A local addition on MM consists of a vector bundle π:EM\pi \colon E \to M, an open neighbourhood of the zero section, UEU \subseteq E, and a smooth map η:UM\eta \colon U \to M such that

  1. The composition of η\eta with the zero section of EE is the identity on MM, and
  2. there exists an open neighbourhood, say VV, of the diagonal in M×MM \times M such that the map π×η:UM×M\pi \times \eta \colon U \to M \times M is a diffeomorphism onto VV.

Remarks

  1. By considering the derivative of η\eta, one can see that the bundle EE must be isomorphic to TMT M. However, it is not necessary to specify an isomorphism in advance since η\eta naturally defines one. Nonetheless, it is common simply to take E=TME = T M.

  2. Another way to simplify the definition is to take U=EU = E. In KM §42.4, this is called a globally defined local addition (though such a conjunction of “global” and “local” may grate). Given a local addition with arbitrary UU, it is possible to define a “globally defined” one simply by using a smooth fibrewise embedding (preserving the zero section) of EE onto an open subset of UU.

  3. A useful source of local additions is from Riemannian geometry: the exponential map coming from a Riemannian structure defines a local addition.

  4. Another source of local additions is to apply the tubular neighbourhood theorem to the embedding of the diagonal MM×MM \to M \times M.

Charts from Local Additions

Charts for MM

Local additions are used in constructing the manifold structure on certain mapping spaces. In brief, if MM is a manifold (possibly infinite dimensional) admitting a local addition, and if NN is a compact manifold (possibly with boundary) then C (N,M)C^\infty(N,M) can be given the structure of a smooth manifold and the construction of the charts uses the local addition on MM. For details, see KM §42. Applying this to the case N=ptN = pt, we obtain charts for MM itself. These charts are useful since then we have charts on both MM and the various mapping spaces which are all defined using the same method and so the relationships between them are that much clearer.

Proposition

Let MM be a smooth manifold, η:EUM\eta \colon E \supseteq U \to M a local addition on MM. Let VM×MV \subseteq M \times M be the image of π×η\pi \times \eta. Let pMp \in M. Let U pE pUU_p \coloneqq E_p \cap U be the fibre of UU over pp. Let V pMV_p \subseteq M be such that {p}×V p=V({p}×M)\{p\} \times V_p = V \cap \left(\{p\} \times M\right). Then the restriction of η\eta to U pU_p defines a diffeomorphism η p:U pV p\eta_p \colon U_p \to V_p.

Proof

Let us start by showing that η p\eta_p is well-defined. Clearly, we can restrict η\eta to U pU_p to get something, the only question is its image. To find that, we look at the image of U pU_p under π×η\pi \times \eta. Since U pE p=π 1(p)U_p \subseteq E_p = \pi^{-1}(p), the image of U pU_p is contained in V({p}×M)V \cap \left(\{p\} \times M\right), which is what we have called V pV_p. Conversely, if qV pq \in V_p then (p,q)V(p,q) \in V and so (π×η) 1(p,q)U(\pi \times \eta)^{-1}(p,q) \in U. Since π(π×η) 1(p,q)=p\pi(\pi \times \eta)^{-1}(p,q) = p, we have (π×η) 1(p,q)UE p(\pi \times \eta)^{-1}(p,q) \in U \cap E_p and thus there is some rU pr \in U_p such that η(r)=q\eta(r) = q. Hence the image of η p\eta_p is V pV_p as claimed.

The rest follows from the simple fact that we can also define η p\eta_p as the restriction of the diffeomorphism (π×η):UV(\pi \times \eta) \colon U \to V to the subset U pU_p on the source and {p}×V p\{p\} \times V_p on the target.

Charts for Mapping Spaces

Local additions are used to great effect in constructing charts for mapping spaces.

Let MM be a smooth manifold (possibly infinite dimensional). Let NN be a functionally compact Frölicher space. Let PMP \subseteq M be a submanifold. Let QNQ \subseteq N be a subset. We consider the space C (N,M;Q,P)C^\infty(N,M;Q,P) of smooth maps NMN \to M which map QQ into PP. As a smooth manifold, MM naturally has the structure of a Frölicher space so this mapping space is well-defined.

We assume that the pair (M,P)(M,P) admits local addition. By that, we mean that MM admits a local addition, say η\eta, with the property that it restricts to a local addition on PP. We shall also assume, for simplicity, that the domain of η\eta is TMT M.

Let g:NMg \colon N \to M be a smooth map with g(Q)Pg(Q) \subseteq P. Let E gE_g be the space of sections of g *TMg^* T M with the property that the sections over QQ are constrained to lie in g *TPg^* T P. In more detail, we define g *TMg^* T M in the usual manner:

g *TM{(x,v)N×TM:g(x)=π(v)} g^* T M \coloneqq \{(x,v) \in N \times T M : g(x) = \pi(v)\}

and then take the space of smooth maps f:Ng *TMf \colon N \to g^* T M with the property that the composition Ng *TMNN \to g^* T M \to N is the identity. Within that space, we further restrict to those ff such that the image of the map Qg *TMTMQ \to g^* T M \to T M lies in TPT P.

Although NN could be quite complicated, because TMMT M \to M is a vector bundle, E gE_g is a vector space. Furthermore, by trivialising g *TMg^* T M using a finite number of trivialisations (possible as NN is functionally compact), we can embed E pE_p as a closed subspace of C (N, n)C^\infty(N,\mathbb{R}^n) for some nn. This embedding shows that E pE_p is a convenient vector space, in the sense of Kriegl and Michor.

We define a map for Φ:E gC (N,M;Q,P)\Phi \colon E_g \to C^\infty(N,M;Q,P) as follows. Let fE pf \in E_p. Then ff is a section of g *TMg^* T M and so is a map Ng *TMN \to g^* T M. By the definition of g *TMg^* T M, we can think of ff as a map NN×TMN \to N \times T M which projects to the identity on the first factor. By applying the projection to the second factor, we obtain a map f^:NTM\hat{f} \colon N \to T M. Composing with η\eta produces a map ηf^:NM\eta \circ \hat{f} \colon N \to M. As fE gf \in E_g, the restriction of f^\hat{f} to QQ lands in TPT P, whence ηf^\eta \circ \hat{f} takes QQ into PP. The map fηf^f \mapsto \eta \circ \hat{f} is what we call Φ\Phi.

Let us identify its image. Let VM×MV \subseteq M \times M be the image of the local addition. Define U gC (N,M;Q,P)U_g \subseteq C^\infty(N,M;Q,P) to be the set of those functions hh such that (g,h):NM×M(g,h) \colon N \to M \times M takes values in VV. We claim that the image of Φ\Phi is U gU_g and that Φ\Phi is a bijection E gU gE_g \to U_g.

Let us start with the image. Let hU gh \in U_g. Then (g,h):NM×M(g,h) \colon N \to M \times M takes values in VV, so we can compose with (π×η) 1(\pi \times \eta)^{-1} to get a map hˇ:NTM\check{h} \colon N \to T M. Together with the identity on NN, we get a map NN×TMN \to N \times T M. By construction, πhˇ=g\pi \check{h} = g and so this map ends up in g *TMg^* T M (which has the subspace structure). Again by construction, the projection of this map to NN is the identity and so it is a section of g *TMg^* T M. That it takes QQ to TPT P follows from the fact that η\eta restricts to a local addition on PP, whence as h(Q)Ph(Q) \subseteq P, hˇ(Q)TP\check{h}(Q) \subseteq T P. Hence Φ\Phi is onto. Moreover, this construction yields the inverse of Φ\Phi and so it is a bijection.

Thus we have charts for C (N,M;Q,P)C^\infty(N,M;Q,P). The next step is the transition functions. To prove this in full generality, we assume not just two different functions at which to base our charts, but also two different local additions to define them. This will show that our resulting manifold structure is independent of this choice.

… to be continued

Diffeomorphisms from Local Additions

Another useful construction from a local addition relates to diffeomorphisms. In differential topology, manifolds have a certain amount of “flexibility”: a common picture is of a rubber sheet that can be deformed. The amount of flexibility is less than that allowed in algebraic topology, but certainly more than that in differential geometry. Being able to deform a manifold a little is a very useful trick in establishing the relationships between the various mapping spaces (for example, it is used in showing that the loop space fibration ΩMLMM\Omega M \to L M \to M is a fibre bundle and not just a fibration) and so it is useful to have a consistent way to perform these deformations that is compatible with a given local addition.

The aim of this construction is to produce, for each vUv \in U, a 1-parameter family of diffeomorphisms, ϕ (t,v)\phi_{(t,v)}, such that ϕ (t,v)(π(v))=η(tv)\phi_{(t,v)}(\pi(v)) = \eta(t v). That is, the basepoint p=π(v)p = \pi(v) is deformed along the path η(tv)\eta(t v) to η(v)\eta(v), and the rest of the manifold is dragged along behind it. (In actual fact, only a very small amount of the manifold is dragged along - although our manifolds are flexible, they have a high amount of tension.)

So let η:UM\eta \colon U \to M be a local addition. Let ν:U\nu \colon U \to \mathbb{R} be a smooth function with the following fibrewise property: for each rr \in \mathbb{R}, the set {qU p:ν(q)r}\{q \in U_p : \nu(q) \le r\} is a compact, radial subset. The square of the norm coming from a smooth orthogonal structure would suffice. Let ρ:[0,1]\rho \colon \mathbb{R} \to [0,1] be a smooth function such that ρ(t)=1\rho(t) = 1 for t0t \le 0 and ρ(t)=0\rho(t) = 0 for t1t \ge 1.

Now for vUv \in U, we define a vector field X^ v\hat{X}_v on U pU_p, where p=π(v)p = \pi(v), by

X^ v(w)ρ(ν(w)ν(v))v \hat{X}_v(w) \coloneqq \rho(\nu(w) - \nu(v))v

Notice that if ν(w)ν(v)+1\nu(w) \ge \nu(v) + 1 then X^ v(w)=0\hat{X}_v(w) = 0 whilst if ν(w)ν(v)\nu(w) \le \nu(v) (so in particular if w=tvw = t v for some t[0,1]t \in [0,1]) then X^ v(w)=v\hat{X}_v(w) = v. By the first of these, we see that X^ v\hat{X}_v is compactly supported on U pU_p. We can transfer it via η p\eta_p to V pV_p and, as it is compactly supported, extend it to all of MM by defining it to be 00 outside V pV_p. Let us write X vX_v for this vector field.

By construction, the map vX vv \to X_v is smooth. As X vX_v is compactly supported, we can apply the exponentiation map 𝒳 c(M)Diff(M)\mathcal{X}_c(M) \to Diff(M) and so obtain, for each vUv \in U, a 1-parameter family of diffeomorphisms of MM, say ϕ (t,v)\phi_{(t,v)}. From the construction of X vX_v, it is clear that all the “action” of this family of diffeomorphisms takes place in V pV_p. Moreover, by construction, the point pp moves along the line tη(tv)t \mapsto \eta(t v) for t[0,1]t \in [0,1].

References

  • Kriegl and Michor, A Convenient Setting of Global Analysis
Revised on April 27, 2010 15:39:20 by Urs Schreiber (131.211.36.96)