Affine Focal Set - Geometric Interpretation

Geometric Interpretation

Let us consider a local parametrisation of M. Let be an open neighbourhood of 0 with coordinates, and let be a smooth parametrisation of M in a neighbourhood of one of its points.

The affine normal vector field will be denoted by . At each point of M it is transverse to the tangent space of M, i.e.

For a fixed the affine normal line to M at may be parametrised by t where

The affine focal set is given geometrically as the infinitesimal intersections of the n-parameter family of affine normal lines. To calculate this we choose an affine normal line, say at point p; then we look at the affine normal lines at points infinitesimally close to p an see if any intersect the one at p. If we choose a point infinitesimally close to, then it may be expressed as where represents the infinitesimal difference. Thus and will be our p and its neighbour.

For t and we try to solve

This can be done by using power series expansions, and is not too difficult; it is lengthy and has thus been omitted.

We recall from the article affine differential geometry that the affine shape operator S is a type (1,1)-tensor field on M, and is given by, where D is the covariant derivative on real (n + 1)-space (for those well read: it is the usual flat and torsion free connexion).

We find that the solutions to are when 1/t is an eigenvalue of S and that is a corresponding eigenvector. The eigenvalues of S are not always distinct: there may be repeated roots, there may be complex roots, and S may not always be diagonalisable. For, where denotes the greatest integer function, there will generically be (n − 2k)-pieces of the affine focal set above each point p. The −2k corresponds to pairs of eigenvalues becoming complex (like the solution to as a changes from negative to positive).

The affine focal set need not be made up of smooth hypersurfaces. In fact, for a generic hypersurface M, the affine focal set will have singularities. The singularities could be found by calculation, but that may be difficult, and we still have no idea of what the singularity looks like up to diffeomorphism. If we use some singularity theory then we get much more information.