Stratified affine spaces

These are quick notes for a joint work with Ch. Wacheux

Here, by an affine manifold, we mean a paracompact manifold with a locally flat affine structure. If the manifold is of dimension n, then near each point we have a chart whose coordinate functions are affine functions, and any local affine function is a linear combination of these functions plus a constant. The differential of a affine function will be called a constant 1-form. The space of local constant 1-forms near each point is a vector space of dimension n. The sheaf of local constant 1-forms is a locally free Abelian sheaf of rank n, i.e. it’s the seaf of local horizontal sections of a vector bundle of rank n with a flat (Gauss-Manin) connection.

Stratified Spaces:

By a stratified  space, we mean a Hausdorff topological space B, which is a disjoint union

B = \union_i B_i

such that:

1) Each B_i is a manifold (called a stratum of B)

2) The boundary of each B_i in B is a locally finite union of strata of lower dimensions than the dimension of B_i.

3) (Topologial splitting property) For each point x in B there is a neighborhood U(x) of x, which intersects with only a finite number is strata, and  which is homeomorphic to a direct product

D^k \times C

where k is the dimension of the D^k is a k-dimensional disk, and C is a stratified space

(This last condition is a deductive condition for each singular point, similar to the local splitting condition for singular foliations)

Stratified Affine Spaces

By a stratified affine space, we mean a stratified space, which satisfies the following additional condition:

4) B is equipped with a sheaf of local functions, called the sheaf of local affine functions of B,  such that the restriction of these functions to each stratum gives rise to a well-defined affine structure on the stratum. Moreover,  for each point x in B of rank k (i.e. the dimension of the stratum containing x is k),  there exist k local affine functions f_1, …, f_k in a neigborhood U(x) of x, such that the restrictions of these functions to each stratum are affinely independent affine functions (i.e. no nontrivial linear combination plus a constant is identically zero), and such that the joint level sets {f_1 = const., …, f_k = const} of these functions can appear as {point} \times C in the local topological splitting condition 3).

The differential of a local affine function on a stratified space is still called a constant 1-form.

Condition 4 implies in particular that the stalk of the sheaf of constant 1-forms at a point x of rank k is a vector space of dimension at least k (but can be greater than k).

Examples:

– Base space of proper integrable Hamiltonian systems

– Affine manifold with boundary?

Maps and subspaces

A map \phi from a stratified space B to a stratified space C is called an affine map if the pull-back of every local affine function (constant 1-form) on C is an affine function (constant 1-form) on B, and moreover each local function on the image \phi(B) whose pull-back is an affine function can be extended to a local affine function on C.

In particular, a topological subspace is called a affine stratified subspace if it’s a stratfied subspace with the induced topology, and it can be equipped with a stratified affine structure such that the inclusion map is an affine map. Notice that the affine structure on the subspace is then unique and induced from the affine structure on the ambient space.

Affine geodesics

An affine geodesic in B is the image of an affine map from a interval (with the standard affine structure) to B.

Star-shaped

B is called star-shaped with respect to a regular point x if the set of points in B which can be connected to x by an affine geodesic contains an open dense subset of B.

Convexity

B is called (intrinsically) convex if it is star-shaped with respect to every regular point.

B is called locally convex if every point damits a neighborhood which is convex.