On a

Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...

, certain curves are distinguished as

timelike In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...

. A timelike homotopy between two timelike curves is a

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

such that each intermediate curve is timelike. No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves (or timelike multiply connected). A manifold such as the

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

can be

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...

(by any type of curve), and at the same time be timelike multiply connected. Equivalence classes of timelike homotopic curves define their own fundamental group, as noted by Smith (1967). A smooth topological feature which prevents a CTC from being deformed to a point may be called a timelike topological feature.

References

* * Algebraic topology Homotopy theory Lorentzian manifolds {{topology-stub