Yes. Note that by composing with tanh, it suffices to find some f : (-1, 1) --> R^3 with these properties. We'll construct such an example with codomain R^2 (which embeds into R^3). Firstly, we take the 'bump function': b : (0, 1) --> R b(x) = exp(1/(x-1) - 1/x) which has the property that at both 0 and 1, it and all of its derivatives are 0. We now integrate it twice and normalise to give a C^infinity function: g : (0, 1) --> R which has g'(0) = 0, g'(1) = 1, and all further derivatives are zero at both endpoints. We can take the graph of this function and reflect in the normal to the point (1, g(1)) to obtain some curve in R^2 (a 'rounded corner') which has these properties: (a) The endpoints are (0, 0) and (c, c), where c = 1 + g(1). (b) The image of the curve is a subset of [0, c]^2, and only the endpoints lie on the boundary of this square. (c) The curve is C^infinity. Now, we can assemble rounded corners and line segments like so: /---v---\ |...|...| \---^---/ The resulting set is homeomorphic to the graph K_(3, 2). We now define, in the obvious way, a Eulerian path beginning at one of the degree-3 vertices and ending at the other, parametrised by arc-length. Now this is a C^infinity injection from (0, l) to R^2 with compact image, and we can compose with tanh and some boring linear polynomial to get a similar injection from R to R^2. Best wishes, Adam P. Goucher
Sent: Wednesday, February 10, 2016 at 7:57 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Does such an curve exist in 3-space?
Let
f: R —> R^3
be a C^oo one-to-one mapping of the reals into 3-space.
(For convenience, assume WLOG that ||f'(t)|| is never 0.)
Suppose further that
a) The image f(R) is closed and bounded in R^3;
and
b) If for some sequence t_j in R we have
lim f(t_j) = f(t) j—>oo
for some t in R, then we also have convergence of the tangent vectors:
lim f'(t_j) = f'(t) j—>oo
-------------------------------------------------
Question: Does there exist such a curve as f ???
(Note: If not for b), an easy example would be for the image f(R) to be a figure-8.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun