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A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

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 π 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 ( A ) < ∞ . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle ( V , v ) | V ∈ G ( n , m ) , v ∈ V such that, for all P ∈ A, one has m ( n - m ) ( V ∈ G ( n , m ) | ( V , π V ( P ) ) ∈ Z ) > 0 . One can replace “for all P ∈ A” by “for m -a.e. P ∈...

A new approach for describing instantaneous line congruence

Rashad A. Abdel-Baky, Ashwaq J. Al-Bokhary (2008)

Archivum Mathematicum

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...

A new characterization of the sphere in R 3

Thomas Hasanis (1980)

Annales Polonici Mathematici

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 √ H K r , for some constants c and r, where H is the mean curvature of M.

A note on surfaces with radially symmetric nonpositive Gaussian curvature

Joseph Shomberg (2005)

Mathematica Bohemica

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...

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