# 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 $M$ and $M'$ be metric spaces, $f,g:M\to M'$ two maps, and $a\in M$.

###### Definition

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

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

is continuous at $x=a$.

Now let $a'\in M'$ be another point.

###### Definition

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

## References

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