Focal stabilty of Riemann metrics
Charles Pugh | Kupka, Ivan | Peixoto, Mauricio
Geodesics; Focal Decomposition; Focal Stability
Let M be a complete Riemann manifold of dimension m and let R* be the space of all complete Riemann metrics on M. If p is a point in M we will consider the totality of geodesics emanating from p and we will be interested in how many of these geodesics will meet again, i.e. focus afterwards at some point q after describing geodesic paths of the same length from p to q. The focal index of a vector v belonging to T(p)M, the tangent space of M at p, is the cardinality of all vectors w in T(p) with the same length as v, |w| = |v|, exp(w) = exp(v). If we call s(i) the totality of all vectors of T(p)M with focal index I, then T(p)M is the union of all s(i), i = 1, 2, Ö and this is called the FOCAL DECOMPOSITION of M with respect to the base point p. Now if S(i) stands for the union of s(i) relative to all p in M, we get a partition of the tangent bundle TM that is the FOCAL DECOMPOSITION of M. Of course Focal Decomposition depends only on the Riemann metric on M. It is also a global concept: all geodesic through p play a role in the construction of the sets s(i), and all geodesics in M play a role in the construction S(i). Fixed a metric g in M, the pointwise index of g at p in M, I(g,p), is defined as the largest i for which s(i,p) is non empty; the uniform index of g, I(g), is defined as the largest i S(i,g) is non empty. Below we state the main results of this paper. POINTWISE INDEX THEOREM: Given p in M there is a residual set G(p) in R* such that, for all g in G(p), I(g,p) is equal to or less than m+1. UNIFORM INDEX THEOREM: If M is compact then there is a residual ser G in R* such that for all g in G, I(g) is equal to or less than 2m+2. In the paper we also give a natural definition of focal stability and state the FOCAL STABILITY CONJECTURE: Given p in M the generic Riemann metric in M is focally stable. This is verified in several specific cases.