mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a product metric is a

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

on the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of finitely many

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...

(X_1,d_),\ldots,(X_n,d_)

which metrizes the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...

. The most prominent product metrics are the ''p'' product metrics for a fixed

p\in''p'' norm of the ''n''-vector of the distances measured in ''n'' subspaces:
: d_p((x_1,\ldots,x_n),(y_1,\ldots,y_n)) = \, \left(d_(x_1,y_1), \ldots, d_(x_n,y_n)\right)\, _p For p=\infty this metric is also called the sup metric:
: d_ ((x_1,\ldots,x_n),(y_1,\ldots,y_n)) := \max \left\.

Choice of norm

For Euclidean spaces, using the L₂ norm gives rise to the Euclidean metric in the product space; however, any other choice of ''p'' will lead to a topologically equivalent metric space. In the category of metric spaces (with Lipschitz maps having Lipschitz constant 1), the product (in the category theory sense) uses the sup metric.

The case of Riemannian manifolds

For

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

(M_1,g_1)

and

(M_2,g_2)

, the product metric

g=g_1\oplus g_2

M_1\times M_2

is defined by :

g(X_1+X_2,Y_1+Y_2)=g_1(X_1,Y_1)+g_2(X_2,Y_2)

for

X_i,Y_i\in T_M_i

under the natural identification

T_(M_1\times M_2)=T_M_1\oplus T_M_2

References

*. * . {{DEFAULTSORT:Product Metric Metric geometry