Curvature in Differential Geometry

Charles G. Gunn, Tim Hoffmann, Ulrich Pinkall, Nicholas Schmitt

Description


In differential geometry we aim to examine curved shapes or spaces, e.g. space curves or surfaces — even in higher dimensions.
Common questions that arise in this context are: „What is the difference in the curvature of an egg, or a cooling tower?“ and „How can we measure it?“.

Of particular interest are surfaces that are „optimal“ in some sense, i.e. surfaces for which a certain measure of quality cannot be improved by slightly perturbing the surface.

A common example are soap skins spanned into a wireframe. Neglecting effects of gravity, these will form surfaces of minimal area which have the boundary prescribed by the wireframe. On the other hand, soap bubbles minimise their surface area under all surfaces that enclose a fixed volume of air.

From the differential geometric viewpoint both examples have the distinguished property of a constant „mean curvature“.

Shown are examples of surfaces of constant mean curvature (helicoid with handles, nodoid, tetranoid, twizzle-torus, Björling-surface, discrete minimal surface) and minimal total mean curvature (Boy’s surface). Also there is one example of a curved three-dimensional space with constant curvature (hyperbolic 3-space).

These and more images can also be seen at imaginary.org .

References


  • Werner Boy.
    Über die Curvatura integra und die Topologie geschlossener Flächen.
    Mathematische Annalen, 57(2):151–184, June 1903.
    doi:10.1007/BF01444342.
  • Hermann Karcher and Ulrich Pinkall.
    Die Boysehe Fläche in Oberwolfach.
    Mitteilungen der DMV, Volume 5 (1), March 1997.
    dgd:549.

Prof. Dr. Tim Hoffmann   +

Projects: A02
University: TU München
E-Mail: hoffmant[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/TimHoffmann


Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin
E-Mail: pinkall[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~pinkall/


Dr. Nicholas Schmitt   +

University: TU Berlin
Website: http://page.math.tu-berlin.de/~schmitt/