At its simplest, a smooth manifold is a place where one can do calculus without worries. Let us consider the formula for a derivative:
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.
The broadest definition is the following.
A local addition on consists of a vector bundle , an open neighbourhood of the zero section, , and a smooth map such that
By considering the derivative of , one can see that the bundle must be isomorphic to . However, it is not necessary to specify an isomorphism in advance since naturally defines one. Nonetheless, it is common simply to take .
Another way to simplify the definition is to take . 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 , it is possible to define a “globally defined” one simply by using a smooth fibrewise embedding (preserving the zero section) of onto an open subset of .
A useful source of local additions is from Riemannian geometry: the exponential map coming from a Riemannian structure defines a local addition.
Another source of local additions is to apply the tubular neighbourhood theorem to the embedding of the diagonal .
Local additions are used in constructing the manifold structure on certain mapping spaces. In brief, if is a manifold (possibly infinite dimensional) admitting a local addition, and if is a compact manifold (possibly with boundary) then can be given the structure of a smooth manifold and the construction of the charts uses the local addition on . For details, see KM §42. Applying this to the case , we obtain charts for itself. These charts are useful since then we have charts on both and the various mapping spaces which are all defined using the same method and so the relationships between them are that much clearer.
Let be a smooth manifold, a local addition on . Let be the image of . Let . Let be the fibre of over . Let be such that . Then the restriction of to defines a diffeomorphism .
Let us start by showing that is well-defined. Clearly, we can restrict to to get something, the only question is its image. To find that, we look at the image of under . Since , the image of is contained in , which is what we have called . Conversely, if then and so . Since , we have and thus there is some such that . Hence the image of is as claimed.
The rest follows from the simple fact that we can also define as the restriction of the diffeomorphism to the subset on the source and on the target.
Local additions are used to great effect in constructing charts for mapping spaces.
Let be a smooth manifold (possibly infinite dimensional). Let be a functionally compact Frölicher space. Let be a submanifold. Let be a subset. We consider the space of smooth maps which map into . As a smooth manifold, naturally has the structure of a Frölicher space so this mapping space is well-defined.
We assume that the pair admits local addition. By that, we mean that admits a local addition, say , with the property that it restricts to a local addition on . We shall also assume, for simplicity, that the domain of is .
Let be a smooth map with . Let be the space of sections of with the property that the sections over are constrained to lie in . In more detail, we define in the usual manner:
and then take the space of smooth maps with the property that the composition is the identity. Within that space, we further restrict to those such that the image of the map lies in .
Although could be quite complicated, because is a vector bundle, is a vector space. Furthermore, by trivialising using a finite number of trivialisations (possible as is functionally compact), we can embed as a closed subspace of for some . This embedding shows that is a convenient vector space, in the sense of Kriegl and Michor.
We define a map for as follows. Let . Then is a section of and so is a map . By the definition of , we can think of as a map which projects to the identity on the first factor. By applying the projection to the second factor, we obtain a map . Composing with produces a map . As , the restriction of to lands in , whence takes into . The map is what we call .
Let us identify its image. Let be the image of the local addition. Define to be the set of those functions such that takes values in . We claim that the image of is and that is a bijection .
Let us start with the image. Let . Then takes values in , so we can compose with to get a map . Together with the identity on , we get a map . By construction, and so this map ends up in (which has the subspace structure). Again by construction, the projection of this map to is the identity and so it is a section of . That it takes to follows from the fact that restricts to a local addition on , whence as , . Hence is onto. Moreover, this construction yields the inverse of and so it is a bijection.
Thus we have charts for . 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 …
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 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 , a 1-parameter family of diffeomorphisms, , such that . That is, the basepoint is deformed along the path to , 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 be a local addition. Let be a smooth function with the following fibrewise property: for each , the set is a compact, radial subset. The square of the norm coming from a smooth orthogonal structure would suffice. Let be a smooth function such that for and for .
Now for , we define a vector field on , where , by
Notice that if then whilst if (so in particular if for some ) then . By the first of these, we see that is compactly supported on . We can transfer it via to and, as it is compactly supported, extend it to all of by defining it to be outside . Let us write for this vector field.
By construction, the map is smooth. As is compactly supported, we can apply the exponentiation map and so obtain, for each , a 1-parameter family of diffeomorphisms of , say . From the construction of , it is clear that all the “action” of this family of diffeomorphisms takes place in . Moreover, by construction, the point moves along the line for .