Curvature and the Shape Operator

Metric Properties and the Fundamental Forms

Our real goal is thus to create a conformal map of the brain's surface. This type of map is unique because it maintains what are called the metric properties of the surface it is mapping.

Metric properties are properties of surfaces that are stable provided that any deformation of that the surface undergoes does not affect the distance between any two points on that surface. Properties that are considered metric properties are distance, area, and curvature.

These are, of course, the most basic and fundamental properties of a surface and this is why mathematical methods used to calculate them are called fundamental.

We can see from this example that the distance between two points on a non-flat surface is a difficult property to maintain if we try to flatten it. Various attempts to do so over the centuries have given us a multitude of different maps of the world in which we live, each one emplying a different method for maintaining these distances.

Maps that are able to maintain the distances between any two points on a surface are called conformal maps.