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Given a smooth regular curve $\gamma$ in $\mathbb{R}^{3}$, one defines the tangent indicatrix of $\gamma$ to be the spherical curve $\gamma'/\lVert \gamma'\rVert$. It is then natural to look for spher …

Given an immersed submanifold $M$ of a Riemannian manifold $\overline{M}$, the first normal space of $M$ at a point $p \in M$ is defined as the linear subspace $N_{p}^{1}M$ of $N_{p}M$ spanned by the …

OK I got confused really bad yesterday. The set $ \{\xi \in N_{t}\gamma \mid A_{\xi}=0 \}$ is indeed $(N-1)$-dimensional. Without loss of generality, assume $\gamma$ be unit-speed, and $\overline{D}_{ …

This is a cross-post from MSE (https://math.stackexchange.com/q/4330772/242708). I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some con …

Following Anton Petrunin’s suggestion, I revise the question to make it less vague. Let $M^{m}$ be an $m$-dimensional Riemannian manifold, and let $\gamma$ be a unit-speed curve $I \to M^{m}$. We say …

A smooth $m$-dimensional submanifold of $\mathbb{R}^{d}$ is said to be $k$-ruled if it is foliated by $k$-dimensional planes, called rulings. Let $M$ be a $k$-ruled submanifold. Then $M$ can be parame …

Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property: For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of …

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form $$\begin{align} \sigma \colon I …

I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$. I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, …

Let $M$ be a submanifold of a Riemannian manifold $Q$, and let $\Gamma$ be a $d$-dimensional submanifold of $M$, i.e., $\Gamma^{d} \subset M \subset Q$. Suppose that, for all (unit) normal vectors of …