Tangentenraumpunkt

A Tangentenraumpunkt, which translates to "tangent space point" in English, is a concept primarily used in differential geometry and related fields. It represents a point within the tangent space of a manifold at a specific point. To understand this, we first consider a manifold, which is a mathematical space that locally resembles Euclidean space. For example, a sphere is a manifold; at any small patch, it looks like a flat plane.

The tangent space at a point on a manifold is a vector space that captures the "infinitesimal" directions in which one can move from that point. Imagine standing on the surface of the Earth at a specific location. The tangent space at that point would be a flat plane representing all possible directions you could start walking in from that exact spot.

A Tangentenraumpunkt is then a specific vector within this tangent space. It signifies a particular velocity or direction of movement emanating from the original point on the manifold. For instance, if the manifold is a curve, the tangent space at a point on the curve is a line, and a Tangentenraumpunkt would be a vector on that line indicating a direction and speed along the curve. This concept is fundamental for defining derivatives of functions on manifolds and understanding the local behavior of geometric objects.