torsionsfri

Torsionsfri, or torsion-free, is a term used in several areas of mathematics to describe the absence of torsion. In an additive group, an element has torsion if some nonzero integer multiple of it is zero. A group is torsionsfri if the only element of finite order is the identity. This idea extends to modules over a domain: a module M is torsionsfri if, whenever r is a nonzero element of the ring and rm = 0, then m = 0. Equivalently, no nonzero element is annihilated by a nonzero ring element.

In differential geometry, a connection on a differentiable manifold is torsionsfri if its torsion tensor T(X,Y) = ∇_X Y − ∇_Y X − [X,Y] vanishes for all vector fields X and Y. The Levi-Civita connection on a Riemannian manifold is a canonical example of a torsion-free connection, reflecting the compatibility with the metric and the symmetry of covariant differentiation.

Examples in group theory and linear algebra include the additive group of integers Z, the group of rational numbers Q, and the free abelian group Z^n; these are torsionsfri. In contrast, any finite group or a group with an element of finite order is not torsionsfri.

In algebraic geometry and commutative algebra, a coherent sheaf or a module is torsionsfri if it has no elements annihilated by a nonzero divisor; for modules over a domain, torsion-free modules generalize submodules of vector spaces. Vector bundles are torsionsfri and, more generally, torsion-free sheaves are locally free on a dense open subset.

Overall, the torsionsfri property signals a form of rigidity: elements behave continuously under scaling or differentiation, without finite-order obstructions.