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As asked, the answer to the question is 'no'. The simply-connected cover $f:\mathbb{R}^2\to S$ of Sherck's first surface $S$ (which is defined in $\mathbb{R}^3$ by the equation $\mathrm{e}^{z} \cos x …

The answer is already 'no' in the first nontrivial case: A 3-manifold in $\mathbb{R}^d$ (where $d>3$) that is ruled by lines (i.e., $k=1$). One can see this as follows: As the OP notes, one can writ …

The question is essentially equivalent to the following classical question: Given a smooth surface $S$ and a bundle map $L:TS\to TS$, when does there exist an immersion $x:S\to \mathbb{R}^3$ such tha …

You can always do this, but it's not as simple as using some kind of ODE (such a flow of vector fields) to construct such charts. First, assume that your surface is connected and simply-connected. Th …

Here are some comments about the OP's question that don't give a definitive answer to the final question (although the answer may well be 'no', see below), but do provide more information, at least in …

Note: I'm revising my answer to make the argument/construction more transparent. In the previous version, I stated an existence result about flat surfaces, but didn't indicate a proof (because, at th …

There are lots of finite dimensional curve families that are closed under Minkowski sum with circular disks. Here's a way to construct examples: First, choose a finite dimensional space $\mathcal{C}$ …

Your question is not very clearly phrased, which may explain why you didn't get any answers on MSE. When you say, "spherical curve $\gamma$ whose tangent indicatrix is the same as the original curve, …

This is sort of an answer and sort of not. I'll let you be the judge: Suppose you formulate the question, not in terms of 'motion' (which you left vague) but terms of 'freely copying' a triangle $T$ …

There is a straightforward way to deduce necessary conditions for a space curve to lie on an ellipsoid, and it's really a matter of calculation to make these conditions explicit in terms of the curvat …

I'm rearranging my answer a little bit because I realized that I overlooked an apparent possibility (that turns out not to occur), and I didn't want my answer to be misleading: If the surface in Eucl …

You'll find a discussion of the analysis of linear Weingarten surfaces via exterior differential systems in these lecture notes of mine. Particularly look at Section 5.1, where it is discussed at len …

Setting aside the assumption that $\phi$ be a polynomial mapping for the moment (however, see below for a construction of a large family of polynomial solutions), if one makes the 'nondegeneracy' assu …

Because people have asked for it, I thought I would supply an example of what I mentioned in my comment above, an immersion of the $3$-sphere into $\mathbb{R}^4$ that has three distinct principal curv …

As others have pointed out, it's not hard to show that any function $F(\kappa_1,\kappa_2)$ that is intrinsic to the surface metric must be a function of $K = \kappa_1\kappa_2$, so that settles what on …