# Metric differential

Metric differential

In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademarcher's theorem to metric space-valued Lipschitz functions.

## Discussion

Rademacher's theorem states that a Lipschitz map f : Rn → Rm is differentiable amost everywhere in Rn; in other words, for almost every x, f is approximately linear when you look in a small enough neighborhood of x. If f is a function from a Euclidean space Rn that takes values instead in a metric space X, it doesn't immediately make sense to talk about differentiability since X has no linear structure a priori. Even if you assume that X is a Banach space and ask whether a Fréchet derivative exists almost everywhere, this does not hold. For example, consider the function f : [0,1] → L1([0,1]), mapping the unit interval into the space of integrable functions, defined by f(x) = χ[0,x], this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ x ≤ y≤ 1, then

$|f(x)-f(y)|=\int_0^1 |\chi_{[0,x]}(t)-\chi_{[0,y]}(t)|\,dt = \int_x^y \, dt = |x-y|,$

but one can verify that limh→0(f(x + h) −  f(x))/h does not converge to an L1 function for any x in [0,1], so it is not differentiable anywhere.

However, if you look at Rademacher's theorem as a statement about how a Lipschitz function stabilizes as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of f instead of its linear properties.

## Definition and existence of the metric differential

A substitute for a derivative of f:Rn → X is the metric differential of f at a point z in Rmn which is a function on Rm defined by the limit

$MD(f,z)(x)=\lim_{r\rightarrow 0} \frac{d_{X}(f(z+rx),f(z))}{r}$

whenever the limit exists (here d X denotes the metric on X).

A theorem due to Bernd Kirchheim[1] states that a Rademacher theorem in terms of metric differentials holds: for almost every z in Rm, MD(fz) is a seminorm and

$d_X(f(x),f(y)) - MD(f,z)(x-y) = o(|x-z|+|y-z|). \,$

What this little-o notation means is that, at values very close to z, the function f is approximately an isometry from Rn with respect to the seminorm MD(fz) into the metric space X.

## References

1. ^ Kirchheim, Bernd (1994). "Rectifiable metric spaces: local structure and regularity of the Hausdorff measure". Proc. of the Am. Math. Soc. 121: 113–124.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia

• Differential geometry — A triangle immersed in a saddle shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines. Differential geometry is a mathematical discipline that uses the techniques of differential and integral calculus, as well as… …   Wikipedia

• Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia

• Metric (mathematics) — In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric …   Wikipedia

• Differential form — In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better[further explanation needed] definition… …   Wikipedia

• Metric tensor (general relativity) — This article is about metrics in general relativity. For a discussion of metrics in general, see metric tensor. Metric tensor of spacetime in general relativity written as a matrix. In general relativity, the metric tensor (or simply, the metric) …   Wikipedia

• Metric (vector bundle) — In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle. Specifically, if M is a topological manifold and E → M a vector bundle on M, then a metric (sometimes called a bundle metric, or fibre metric)… …   Wikipedia

• Metric signature — The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is… …   Wikipedia

• Differential topology — In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.… …   Wikipedia

• Metric derivative — In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of speed or absolute velocity to spaces which have a notion of distance (i.e. metric spaces) but not… …   Wikipedia