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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 Poi …

Here is a general way to construct a large family of bendings of the cylinder $M$, with nonplanar boundaries, via Alexandrov's isometric embedding theorem. All these examples will be convex. First no …

This has been established in a recent paper by Micha Wasem: h-principle for curves with prescribed curvature. Geom. Dedicata 184 (2016), 135–142. In particular Wasem showed that one can prescribe c …

The answer to your question depends on whether you are looking for a tubular neighborhood in the general differential topological sense or the more restrictive geometric sense. The answer in the topol …

How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not kn …

Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in x …

Yes, every closed orientable surface can be embedded in $R^3$ with positive mean curvature. One way to construct higher genus examples is by gluing thin tori, which are mean convex. For instance we ca …

The conjecture attributed to Hadamard, if one regards that as being concerned with the existence of a complete embedded negatively curved surface in a ball, is still open. I have read the correspondin …

The best way I know to illustrate the notion of curvature is via Toponogov's theorem. We can compare any (geodesic) triangle in a Riemannian manifold $M$ with one with the same edge lengths in Euclid …

This is just a bit more elaboration on Anton's nice example, and subsequent comment. Here is the picture of the two triangles: Note that $p'x'y'$ is obtained from $pxy$ by rotating the side $\overlin …

Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian …

It sounds like, to some extent, what you are asking is whether an immersed closed curve in the plane can be extended to an immersion of a disk. This would yield the multiplicities that you seek; howev …

It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of …

Answer to question 1: If a convex surface is not closed, then generally it is far from rigid as it might admit infinitely many isometric deformations; however, if the surface has $\mathcal{C}^{2,1}$ r …

If the high school students are taking, or have already taken, Calculus I, and know how to differentiate, then I would use the notion of acceleration. I would start by asking: what is a straight line …