nLab metric jet

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

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