Faculty of Engineering
Geometry Understood Using Curvature and Metrics
When we look around us, we do not really see that the earth is round. Yet, it is easy to find out on a larger scale. Let us take a triangle on the surface of the earth (without hills and valleys) and sum up the angles between its edges. If we measure carefully enough (or the triangle is very big), we see that the sum of the angles is strictly greater than 180°. Did we not learn in school that the sum is always 180°? Indeed, this is the case for all triangles in the plane. Ergo, the surface of the earth cannot be flat like a plane (beside of the hills and valleys). To be more general, we see that the geometry of a sphere, the surface of a ball, is encoded in parameters as the sum of angles of triangles. The mathematical concept behind this is curvature. The plane has no curvature, we call it flat. The sphere has a curvature. Surfaces like spheres are called elliptic. Their so-called Gauß-curvature is positive. Are there also triangles where the sum of the angles is less than 180°? Yes, triangles on surfaces like a saddle have this property (as for the third object in the picture). Such surfaces are called hyperbolic, their Gauß-curvature is negative.
The Gauß-curvature gives a quite comprehensive understanding of the geometry of surfaces. Considering more complicated objects, for instance, of higher dimension, we have to use different tools generalizing the concept of curvature. Interested in the geometry of so-called complex manifolds, Martin Sera and his collaborators from Germany, Sweden and Japan are studying singular Hermitian metrics on holomorphic vector bundles. A Hermitian metric can be seen as a generalization of measuring length and angles. They are very useful to compute parameters representing the manifold's curvature.Unfortunately, this works only for special kinds of metrics which do not exist on all complex manifolds. By considering metrics with singular behaviour, Martin Sera and his collaborators are able to apply such computations to a much wider range of manifolds than considered before.