nLab
metric jet

Metric jets

Idea

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.

Definition

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

Definition

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.

Definition

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.

References

  • Elisabeth Burroni? and Jacques Penon?, “A metric tangential calculus”, TAC.

Revised on July 25, 2013 22:50:11 by Anonymous Coward (216.220.114.135)