APS of manifolds with given structure group APS
From Apstheory
Contents |
[edit] Definition
Let Failed to parse (Can't write to or create math temp directory): G
be a sub-IAPS of the GL IAPS. Then, the APS of manifolds with structure group APS as Failed to parse (Can't write to or create math temp directory): G is defined as the following APS:
- The Failed to parse (Can't write to or create math temp directory): n^{th}
member is the collection (upto equivalence) of all manifolds with reduction of the structure group to Failed to parse (Can't write to or create math temp directory): G_n (here, the specified structure includes not only the manifold but also the reduction of the structure group)
- The block concatenation map is simply the product manifold of the two given manifolds, with the reduced structure gorup also as the product of the corresponding structure groups.
We shall denote this APS of manifolds for Failed to parse (Can't write to or create math temp directory): G
as Failed to parse (Can't write to or create math temp directory): Man(G)
.
[edit] Examples
[edit] Orthogonal IAPS and Riemannian manifolds
- Further information: APS of all Riemannian manifolds
The orthogonal groups over reals form a sub-IAPS of the general linear IAPS. A Riemannian manifold is a differential manifold along with, for every point, a choice of inner product on the tangent space at that point. Since an inner product can be thought of as a choice of coset for the orthogonal group, we can think of the extra structure as specifying a basis at each point, loose upto multiplication by an orthogonal matrix.
Clearly, for a product of Riemannian manifolds, we can consider the inner product defined by taking the two tangent spaces to be mutually orthogonal, and such that the restriction to each tangent space is the inner product already defined on that.
Since the orthogonal IAPS is saturated, it follows that a Riemannian structure provided to the product of two manifolds also defines a Riemannian structure on the individual manifolds.
[edit] Positive-determinant IAPS and oriented manifolds
{{furtherAPS of all oriented manifolds}}
The positive-determinant IAPS, namely the IAPS comprising matrices of positive determinant, is a sub-IAPS of the GL IAPS, although it is clearly not saturated. An 'oriented manifold is a manifold along with a choice of orientation, or equivalenly a manifold where each tangent space comes equipped with a basis that is loose upto multiplication by some matrix with positive determinant.
Clearly, the product manifold of two oriented manifolds also comes with the structure of an oriented manifold, namely taking the orientation element as the block concatenation of the two orientation elements.
[edit] Symplectic IAPS and symplectic manifolds
- Further information: APS of all symplectic manifolds
The symplectic IAPS, namely the IAPS comprising symplectic matrices, is a sub-IAPS of the GL IAPS when diluted by a factor of 2. A symplectic manifold is an even-dimensional manifold along with a choice of basis loose upto multiplication by a symplectic matrix, at each point.
The APS of all symplectic manifolds associates to each Failed to parse (Can't write to or create math temp directory): n
the collection of all symplectic manifolds of dimension Failed to parse (Can't write to or create math temp directory): n and defines block concatenation via the product manifold construction.
[edit] Relation between structure group APS and the manifold APS=
[edit] Sub-IAPS relation=
If Failed to parse (Can't write to or create math temp directory): G
≤ Failed to parse (Can't write to or create math temp directory): H are sub-IAPSes of the GL IAPS, then there is a homomorphism of APSes from Failed to parse (Can't write to or create math temp directory): Man(G) to Failed to parse (Can't write to or create math temp directory): Man(H)
. This homomorphism preserves the underlying topological manifold, and uses reduction of the structure group to Failed to parse (Can't write to or create math temp directory): G_n
to specify a reduction of the structure group to Failed to parse (Can't write to or create math temp directory): H_n
.
The map may not be injective because there may be many different reductions of the structure group to Failed to parse (Can't write to or create math temp directory): G_n
for the same reduction of the structure group to Failed to parse (Can't write to or create math temp directory): H_n
. It is not surjective because for certain reductions of the structure group to Failed to parse (Can't write to or create math temp directory): H_n , there may not exist any further reduction to Failed to parse (Can't write to or create math temp directory): G_n .
[edit] Intersection of IAPSes
If Failed to parse (Can't write to or create math temp directory): G
and Failed to parse (Can't write to or create math temp directory): H are both sub-IAPSes, then providing a reduction of the structure group to both Failed to parse (Can't write to or create math temp directory): G as well as to Failed to parse (Can't write to or create math temp directory): H is equivalent to providing a reduction of the structure group to Failed to parse (Can't write to or create math temp directory): G ∩ Failed to parse (Can't write to or create math temp directory): H
.
