torsionsfrei

Torsionsfrei is a German adjective that translates to "torsion-free" in English. In mathematics, particularly in differential geometry and algebraic geometry, the concept of being torsion-free is crucial for defining and understanding certain mathematical structures.

In differential geometry, a connection on a vector bundle is said to be torsion-free if its torsion tensor is identically zero. The torsion tensor is a measure of how much the connection fails to preserve the commutative property of parallel transport. A torsion-free connection implies that the geodesics defined by the connection behave in a way that is consistent with the underlying manifold's geometry, without any twisting or shearing. For example, the Levi-Civita connection on a Riemannian manifold is always torsion-free.

In the context of modules over a ring, a module is called torsion-free if every non-zero element is not annihilated by any non-zero element of the ring. More formally, for an R-module M, it is torsion-free if for any non-zero element x in M, the annihilator of x, which is the set {r in R | rx = 0}, contains only the zero element of R. This property is particularly important when studying modules over integral domains. In such cases, torsion-free modules are closely related to projective modules.

The term "torsionsfrei" emphasizes the absence of a specific type of "twist" or "rotation" in a mathematical object, leading to more predictable and well-behaved properties.