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Is there a simple smooth closed curve $\gamma$ in $\mathbb{R}^{3}$ such that for all $x,y\in \gamma$ with $x \neq y $, $l_{x}$ and $l_{y}$ are skew lines, where $l_{x} $ and $l_{y}$ are straight lines tangent to $\gamma$ at $x,y$, respectively? What about real analytic case?

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If by "skew" we mean nonparallel and nonintersecting, then the answer is NO. Every smooth closed curve in $\mathbf{R^3}$ has uncountably many pairs of intersecting tangent lines. This follows from Poincare-Hopf index theorem; see

M. Ghomi, Tangent bundle embeddings of manifolds in Euclidean space, Comment. Math. Helv., 81(2006), 259-270.

The lowest dimensional space where a closed curve with nonparallel and nonintersecting tangent lines may be constructed is $\mathbf{R^4}$. A very natural example is given by $$ \mathbf{C}\supset\mathbf{S}^1\ni z\longmapsto (z,z^2)\in \mathbf{C}^2\simeq\mathbf{R}^4, $$ as described in the above paper.

On the other hand, if we require the tangent lines only to be nonparallel, then the answer is YES, such curves do indeed exist in $\mathbf{R}^3$. They were first constructed by Beniamino Segre, to disprove a (short-lived) conjecture of Hugo Steinhaus, in 1968:

B. Segre, Global differential properties of closed twisted curves, Rendiconti del Seminario Matematico e Fisico di Milano, December 1968, Volume 38, Issue 1, pp 256-263.

They are not difficult to construct: one starts with a closed spherical curve $T\colon[a,b]\to\mathbf{S}^2$ parametrized by arc length which does not have any self intersections, is disjoint from its antipodal reflection, and contains the origin in the interior of its convex hull. The last property ensures the existence of a (positive) density function $\rho\colon [a,b]\to\mathbf{R^+}$ such that the center of gravity of the weighted curve $\rho T$ is the origin, i.e., $\int_a^b\rho(s)T(s)ds=o$. Then we integrate this curve to obtain: $$ \gamma(t):=\int_0^t\rho(s)T(s)\,ds. $$ Choosing $T$ and $\rho$ to be smooth periodic functions will ensure that $\gamma$ is smooth as well. The resulting curve $\gamma$ will be closed, due to the center of mass condition, and will have no pairs of parallel tangent lines, because its tangential spherical image is $T$ by construction.

The first explicit example of these curves probably appeared in

M. Ghomi, Shadows and convexity of surfaces, Ann. of Math., 155 (2002) 281-293,

where it was used to solve Henry Wente's shadow problem. The example in the above paper is both simple and analytic (it is given by polynomials). Also it lies on a convex surface. Later, Bruce Solomon and I studied these curves, which we called skew loops, some more in

M. Ghomi and B. Solomon, Skew loops and quadric surfaces, Comment. Math. Helv., 77 (2002) 767-782.

where it is shown that the only closed surfaces in $\mathbf{R}^3$ which do not admit any skew loops are ellipsoids. The higher dimensional analogues of these objects are also studied in

M. Ghomi and S. Tabachnikov, Totally skew embeddings of manifolds, Math. Z., 258 (2008), 499-512.

Here the term totally skew means nonparallel and nonintersecting.

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    $\begingroup$ A very similar construction is also used in arxiv.org/abs/1211.3162 to solve a regularity problem for cosmic strings. $\endgroup$ – Willie Wong May 11 '16 at 2:43
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    $\begingroup$ Nice example. But how does one make sure that no two tangents intersect? $\endgroup$ – Sebastian Goette May 11 '16 at 12:01
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    $\begingroup$ @Sebastian Goethe: Sorry, I interpreted "skew" as nonparallel. It is not possible for a closed curve in $R^3$ to have nonintersecting tangent lines, but one can construct these in $R^4$. This is described in the paper with Serge Tabachnikov mentioned above. $\endgroup$ – Mohammad Ghomi May 11 '16 at 12:13
  • $\begingroup$ I just edited my answer above to clarify different interpretations of the term "skew", and corresponding answers in each case. $\endgroup$ – Mohammad Ghomi May 11 '16 at 12:28
  • $\begingroup$ @ Willie Wong: Thank you for this reference. Constructing a curve with a prescribed tangential spherical image is the simplest example of convex integration theory which goes back to Whitney. The center of mass trick to ensure that the curve closes up is also mentioned on p. 168 of the book "Partial differential relations" of Gromov. $\endgroup$ – Mohammad Ghomi May 11 '16 at 15:07

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