Re: PL vs. PL locally flat

"Blake Winter" <mad2physicist@xxxxxxxxx> writes:

>So, in particular, is it possible to have PL locally flat nontrivial
>embeddings of the 2-sphere or torus into 4-space? I assume the answer
>is yes; however, I am having trouble figuring out just how one would do
>it - is there a more precise way of thinking about "locally a product"?
> I am thinking part of my confusion comes from the precise definition
>of locally flat. As I understood it, locally flat means, essentially,
>that the differential metric can be defined

Stop there. In the phrase "locally flat", the word "flat" has
*nothing* to do with metrics. The phrase may seem to have been
badly chosen, but it's standard now.

>and is suitably nonsingular
>at every point on the submanifold (at least, that's how I was used to
>thinking about it in general relativity) - essentially you have to have
>some coordinate system in which the metric at that point becomes the
>identity.

Well, in this case you have some coordinate system (but not a smooth
one--a piecewise linear one) in which the subsurface lies in a
linear subspace of dimension 2 (and then, locally, the whole
space is the product of that subspace with its complement).

>Given this, isn't any cone point going to fail to be locally
>flat?

No (as I'm sure you now see). ... You *can* define "curvature"
on a piecewise-linear surface, in such a way that the Gauss-Bonnet
Theorem (and so forth) still hold--but all the curvature is
concentrated at the vertices, and even a vertex with non-zero
curvature can be locally flat (although a non-locally-flat
vertex can't have 0 curvature, I think, because of the Milnor-
Fary Theorem...but I'm not quite sure of this).

>If so, then isn't any 2-manifold in 3 or 4-space with cone
>points going to fail to be locally flat? I presume the definition of
>"locally a product" is going to relate in some way to this.

Lee Rudolph

.