metric jet

Metric jets


The notions of metric tangency and metric jet are generalizations of notions from differential calculus such as tangent vectors and jet spaces to the setting of arbitrary metric spaces.


Let MM and MM' be metric spaces, f,g:MMf,g:M\to M' two maps, and aMa\in M.


We say that ff and gg are tangent at aa if f(a)=g(a)f(a)=g(a) and the function C a:M +C_a:M\to \mathbb{R}_+ defined by

C a(a)=0C a(x)=d(f(x),g(x))d(x,a)xa C_a(a) = 0 \qquad C_a(x) = \frac{d(f(x),g(x))}{d(x,a)} \forall x\neq a

is continuous at x=ax=a.

Now let aMa'\in M' be another point.


The set of jets from (M,a)(M,a) to (M,a)(M',a') is the quotient set of the set of maps f:MMf:M\to M' which are locally Lipschitz? at aa and satisfy f(a)=af(a)=a' by the equivalence relation of tangency at aa.