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October 4

Name the object

I have a vector bundle $\scriptstyle \pi \,:\,E\,\twoheadrightarrow \,X$ with fibre F. At each point x ∈ X, we have a linear map S_x : F → F which varies smoothly with x. What would we call the whole object S : E → E? I think that each S_x is a type (1,1)-tensor, is that right? If that's true then is S a type (1,1)-tensor field? Any suggestions? — Fly by Night (talk) 13:28, 4 October 2010 (UTC)[reply]

Generalized Metric Space

Has anyone thought of generalizing the definition of a metric space in the following direction?

Let X be a set of points. Instead of considering the distance function d to map pairs of elements of X to nonnegative real numbers, let d map into a set S, equipped with a total order ≤, a minimum element 0, and a binary operation + for which 0 is identity and + respects the order in the sense that whenever a, b, c and d are elements of S with a ≤ c and b ≤ d, we have a+b ≤ c+d. Then copy the axioms of a Metric space over, substituting in S, 0, ≤ and + appropriately.

I can quickly think of ways to add more structure to this; this is an extreme generalization. It would certainly be useful to require + to be associative, for example, or to require that it simply be addition in a totally ordered field. My question is has anyone generalized in this direction, and is there a name for such generalized spaces? --24.27.16.22 (talk) 23:22, 4 October 2010 (UTC)[reply]

Sorry, I haven't seen such a generalization, though something similar has occurred to me. In particular, when constructing the real numbers from equivalence classes of Cauchy sequences of rationals, the distance "metric" on Q must be defined as mapping into Q instead of R, since R hasn't been defined yet. This breaks the usual definition of a metric, but can be easily solved by allowing a form of your generalization.

As you point out the set S must be totally ordered and have 0 as it's minimum. I'd say S should definitely be a group under addition: you should have inverses so that A+b <= A+c iff b <= c, commutativity so the symmetry of the metric respects swapping the order in the triangle inequality, associativity so the triangle inequality may be generalized to stopping off at n points in any order (since you're already commutative) instead of just 1 on your way between two points, and 0 as above. If S is finite, S = {0} would require the space itself to be trivial. Finite |S| > 1 wouldn't work, since 0 < a => 0 < na for all integers n, but a has finite order in the additive group S, so eventually 0 < Na = 0. So, S must be infinite or the metric space is trivial.

As for multiplicative structure... it seems quite desirable to be able to go from a < nb (n an integer, a and b in S) to a/n < b since this is used all the time in proofs. This would require a version of multiplicative inverses: going from A = na to A/n = a bijectively. So we could define multiplication on (S, positive integers) so that it's invertible, commutative (since na = an from commutativity of S under +) while special-casing 0a = a0 = 0. This gives us a somewhat horrific version of a group under multiplication. It would extend naturally to multiplication of S by rational numbers, which of course respects the order axioms, i.e. for rational q and p, q >= p iff aq >= ap, with equality holding only when q = p. We've gotten pretty close to an ordered field by now, but we're not quite there. We would definitely want distributivity of the above multiplication: a(n+m) = an + am clearly; (a+b)n = an + bn from commutativity of +; I think the inverses above should quickly imply this distributivity must extend to all rational multiplication.

Now aq = ap iff q = p means a(q-p) = 0 iff q=p iff q-p = 0 and in particular each aq for distinct rational q must be distinct. Picking an arbitrary e in S with e not 0, {eq} is isomorphic to Q at this point since addition and multiplication in {eq} can clearly be mapped to addition and multiplication in Q. So, in some sense S must contain Q as a "subfield" (quotes since S isn't itself a field and the isomorphism is nonstandard). I believe at this point we could call e "1" to give for any a in S, (eq)a = qa. In this way we could extend the multiplication above to use a subset of S itself, which must respect the standard commutative ring axioms and is invertible since Q is a field. So, it seems a reasonable metric should at the very least contain an "isomorphic" copy of Q and be a sort of mutant field under a restricted multiplication operation, which in a restricted sense is also an ordered field.

At the bare minimum, then, Q is the "smallest" space a metric should map into. Under the continuum hypothesis, R would then be the next "largest" S not possibly isomorphic to Q, though perhaps R under some restricted multiplication would also work. At the least, we're quite close to R, though not there yet. If we were to use a countable S, the set of distances between points would be countable, so the metric space itself couldn't be uncountable while preserving d(a, b) = 0 iff a = b. At least for larger metric spaces, we'd need an uncountable S, which suggests using R (at the minimum) the vast majority of the time, though again we might be able to get away without defining multiplication on all of R.

One important property of R is the least upper bound property--in fact any ordered field with the LUB property is isomorphic to R. That property is quite important for large chunks of analysis. For instance, a sum of rationals getting arbitrarily close to Sqrt(2) in R, when considered in Q, does not converge. A similar example should be constructable for any S without LUB. For calculus, the metric should almost certainly have the LUB and be very, very close to R.

So after thinking about it, perhaps you could get something interesting out of taking S to be...

Countable, though this may degenerate into S = Q
Uncountable, with or without LUB but not fully defining multiplication on S--merely define multiplication on a subset {qe} of S as above.

I imagine (1) could be truly interesting if it doesn't degenerate. It would be somewhat akin to the usual discrete vs. continuous divide and so at least heuristically shows promise. I'd guess (2) would give you large chunks of the usual theory and might give a cute little generalization of metric spaces that doesn't serve much practical use. Of course, without looking into the possibilities much more carefully, who knows? 67.158.43.41 (talk) 15:57, 6 October 2010 (UTC)[reply]

One can think of a(n) (ultra)metric space as an Enriched category - fun to see how. Then generalizations are clear. More details if anyone sees this.John Z (talk) 22:24, 16 October 2010 (UTC)[reply]