Different metrics in different dimensions

[Bob3141592]

TL;DR Summary

Given a space in R_n = R_1 X R_2 X R_3 X R_4 ... can the metric for the R_1 x R_2 subspace be different from the metric for the R_3 X R_4 subspace?

I'm trying to get a handle on how general a space in R_n can be. Part of my motivation is the curled up dimensions physicists talk about. How does one dimension work differently than another dimension? Can one part of the dimensional structure follow one metric and another part follow a different metric?

I rather think it should be possible. That raises questions about the combinations of subspaces. Can R_1 X R_2 be different (say, taxicab geometry) from R_1 X R_3 (say, Euclidean) as long as R_1 X R_3 is consistent (um, somewhere in between maybe)?

Sorry if this is worded poorly, and if it's in an inappropriate folder. And how does one access the proper notation symbols?

Thanks.

 

[fresh_42]

Summary: Given a space in R_n = R_1 X R_2 X R_3 X R_4 ... can the metric for the R_1 x R_2 subspace be different from the metric for the R_3 X R_4 subspace?

I'm trying to get a handle on how general a space in R_n can be. Part of my motivation is the curled up dimensions physicists talk about. How does one dimension work differently than another dimension? Can one part of the dimensional structure follow one metric and another part follow a different metric?

Sure. You can build direct products of different topological spaces.

I rather think it should be possible. That raises questions about the combinations of subspaces. Can R_1 X R_2 be different (say, taxicab geometry) from R_1 X R_3 (say, Euclidean) as long as R_1 X R_3 is consistent (um, somewhere in between maybe)?

Sorry if this is worded poorly, and if it's in an inappropriate folder. And how does one access the proper notation symbols?

Thanks.

Phase spaces are considered In stochastic and physics which cover all possible states, i.e. their description. This leads to different dimensions in the components and thus different units and scales.

The actual question is not whether it can be defined rather what should it be good for, i.e. what do you want to do?

 

[jbriggs444]

Summary: Given a space in R_n = R_1 X R_2 X R_3 X R_4 ... can the metric for the R_1 x R_2 subspace be different from the metric for the R_3 X R_4 subspace?

One example would be the four dimensional space-time we live in. You can pick out a two dimensional space-like slice using x and y coordinates and you can pick out an orthogonal two dimensional Minkowski slice using z and t coordinates.