Krümmungsinvariante

Krümmungsinvariante is a concept in differential geometry that refers to a quantity associated with a manifold or its submanifolds that remains unchanged under certain transformations, typically isometries. In simpler terms, it's a property of the curvature that doesn't change when you bend or stretch the object in specific ways.

For a surface in three-dimensional Euclidean space, the Gaussian curvature is a well-known Krümmungsinvariante. This means that if you can deform a surface without stretching or tearing it (an isometry), its Gaussian curvature at corresponding points will be the same. This has significant implications, for instance, it explains why a flat piece of paper cannot be perfectly mapped onto a sphere without distortion.

More generally, in higher dimensions, Krümmungsinvarianten can involve more complex quantities like the Riemann curvature tensor or its contractions. These invariants provide fundamental information about the intrinsic geometry of a space. The study of Krümmungsinvarianten is crucial for understanding the properties of manifolds, including their topological and geometric characteristics. They play a vital role in areas such as general relativity, where the curvature of spacetime is described by these invariant quantities.