Consider a Pseudo-Riemannian Manifold with signature
$$ (\underbrace{+,\cdots,+}_p,\underbrace{-,\cdots,-}_q) $$
For any positive integers $p$ and $q$. Can this kind of manifold contain closed timelike curves (CTCs)? I know that if $p=1$, then we get a Lorentzian Manifold that can't contain CTCs, but I am interested in the cases where $p>1$.
For $p=1$, CTC's do not exist in Minkowski spacetime. In other $1+3$ spacetimes, in principle they are admitted in the absence of further requirements (like globally hyperbolicity) on the causal structure of the spacetime. They must be present if the spacetime is compact, for instance.
For $p\geq 2$, the answer is obviously YES. Consider a manifold $M$ with metric $g$ with signature (p,q) and $p \geq 2$. In a $p+q$-dimensional neighbourhood $U$ of any point $s\in M$, using the exponential map at $s$ starting from a $p$ dimensional subspace generated by $p$ timelike vectors in $T_sM$, you can construct an embedded $p$-dimensional submanifold $N$ passing through $s$ and whose metric (induced by $g$) has signature $(p,0)$. This means that every vector tangent to a point in $N$, considered as a manifold on its own right, is timelike. In local coordinates on $N$ around $s\in N$, any circle surrounding $s$ is a closed timelike curve.
