Difference between "Live" and "Define"

Question

In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used.

I'm interested to know whether there is a difference between the concepts of "to define" and "to live"?

Thanks for your attention!

Answer 1

Considering Joriki's example

This surface lives in $\mathbb R^3$.

the important thing to be aware of is that this is a informal and deliberately imprecise way of speaking. It should never be used to provide any new information (unless you're more than usually sure that whomever you're speaking to will understand it as you mean it), but can be used to call attention to something you already know, or to suggest a way to visualize what you're speaking about. It certainly has no place in a written-down proof or definition, but might be used in informal motivation and explanations.

In general, the content of "$X$ lives in $Y$" is that $X$ is an element of $Y$, or a subset of $Y$ or a function on $Y$, or sometimes even a function with values in $Y$, (such as a parameterized path in $Y$), or ... something like that. In general it invites you to think of $X$ as something that "takes place" inside $Y$ in some appropriate sense. It's up to the listener to imagine which appropriate sense is meaningful in the context, but there's a strong subtext that the exact technical nature of the relation between $X$ and $Y$ will not be of paramount importance for understanding what comes next.

(In the particular example above, the meaning is almost certainly that the surface is a subset of $\mathbb R^3$ and is equipped with the subset topology (or metric) it inherits from the standard topology (or metric) on $\mathbb R^3$).