Closed timelike curve

Remarks

So a closed timelike curve is a closed curve that is timelike and future-directed at every point. If $p$ lies on a closed timelike curve, then $p \in I^+(p)$, and conversely $p \in I^+(p)$ gives a closed timelike curve through $p$. A spacetime with no closed timelike curve is called chronological.

Example: let $M = \R^4 / \sim$ where $(t, x) \sim (t + 1, x)$, with the flat metric $-dt^2 + \abs{dx}^2$ of Minkowski space, time-oriented by $\partial_t$. The curve $\gamma(s) = [(s, 0)]$, $s \in [0, 1]$, has tangent $\partial_t$ with $g(\partial_t, \partial_t) = -1 < 0$, and $\gamma(0) = \gamma(1)$, so it is a closed timelike curve.

Non-example: in Minkowski space the circle $s \mapsto (0, \cos s, \sin s, 0)$, $s \in [0, 2\pi]$, is closed, but its tangent $v = (0, -\sin s, \cos s, 0)$ has $g(v, v) = 1 > 0$, so it is spacelike, not timelike. In fact Minkowski space has no closed timelike curve, since $t$ strictly increases along every future-directed timelike curve.