### 0-tight surfaces with boundary and the total curvature of curves in surfaces

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi}_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^{m}$-measurable subset of ℝⁿ with ${}^{m}\left(A\right)<\infty $. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi}_{V}\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for ${}^{m}$-a.e. P ∈...

We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $-4{d}^{2}/d{s}^{2}+{\kappa}^{2}\left(s\right)$ with potential given by the curvature of a closed curve.

Based on the E. Study’s map, a new approach describing instantaneous line congruence during the motion of the Darboux frame on a regular non-spherical and non-developable surface, whose parametric curves are lines of curvature, is proposed. Afterward, the pitch of general line congruence is developed and used for deriving necessary and sufficient condition for instantaneous line congruence to be normal. In terms of this, the derived line congruences and their differential geometric invariants were...

Let M be a closed connected surface in ${R}^{3}$ with positive Gaussian curvature K and let ${K}_{I}I$ be the curvature of its second fundamental form. It is shown that M is a sphere if ${K}_{I}I=c\surd H{K}^{r}$, for some constants c and r, where H is the mean curvature of M.

It is easily seen that the graphs of harmonic conjugate functions (the real and imaginary parts of a holomorphic function) have the same nonpositive Gaussian curvature. The converse to this statement is not as simple. Given two graphs with the same nonpositive Gaussian curvature, when can we conclude that the functions generating their graphs are harmonic? In this paper, we show that given a graph with radially symmetric nonpositive Gaussian curvature in a certain form, there are (up to) four families...