Product metric

In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces $(X_{1},d_{X_{1}}),\ldots ,(X_{n},d_{X_{n}})$ which metrizes the product topology. The most prominent product metrics are the p product metrics for a fixed $p\in [1,\infty )$ : It is defined as the p norm of the n-vector of the distances measured in n subspaces:

$d_{p}((x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}))=\|\left(d_{X_{1}}(x_{1},y_{1}),\ldots ,d_{X_{n}}(x_{n},y_{n})\right)\|_{p}$

For $p=\infty$ this metric is also called the sup metric:

$d_{\infty }((x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})):=\max \left\{d_{X_{1}}(x_{1},y_{1}),\ldots ,d_{X_{n}}(x_{n},y_{n})\right\}.$

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 manifolds $(M_{1},g_{1})$ and $(M_{2},g_{2})$, the product metric $g=g_{1}\oplus g_{2}$ on $M_{1}\times M_{2}$ 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 $X_{i},Y_{i}\in T_{p_{i}}M_{i}$ under the natural identification $T_{(p_{1},p_{2})}(M_{1}\times M_{2})=T_{p_{1}}M_{1}\oplus T_{p_{2}}M_{2}$.

References

• Deza, Michel Marie; Deza, Elena (2009), Encyclopedia of Distances, Springer-Verlag, p. 83.
• Lee, John (1997), Riemannian manifolds, Springer Verlag, ISBN 978-0-387-98322-6.