Product metric explained

(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 , inf ty) :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 =inf ty 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 ⊕} g _{2} on M _{1 x} 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 x} M _{2)=T} ^{p _{1}} M _{1 ⊕} T _{p _{2}} M _{2} .

References

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