Pseudo-Riemannian Manifold Without Nonzero Null Vectors – Terminology

Question

Is there a name for a pseudo-Riemannian manifold that admits no nonzero null vectors? More precisely: For a pseudo-Riemannian manifold $(R,g)$, a null vector is a non-zero vector field $X:M \to TM$ such that
$$
g(X,X)(m) = 0, \forall m \in M.
$$
As this [question][1] shows – vector like this exist. But can there exist manifolds where they do not exist and do such manifolds have a name?

Answer 1

Note first that every pseudo-Riemmanian manifold admits a null vector field which is not identically $0$ (just construct one locally and multiply it by a bump function). So by "non-zero vector field" I assume you mean "nowhere vanishing".

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. The tangent bundle $TM$ always admits an orthogonal splitting as $E \overset{\perp}{\oplus} F$, where $E$ and $F$ are respectively positive and negative definite (hence of respective rank $p$ and $q$). Moreover this splitting is unique up to homotopy (because, pointwise, the set of such splittings is the symmetric space of the orthogonal group $O(p,q)$, which is contractible).

Proposition: $M$ admits a nowhere vanishing null vector field if and only if $E$ and $F$ both admit nowhere vanishing sections.

Proof: Decompose a nowhere vanishing null vector field $X$ as $X_E + X_F$. Then $g(X_E,X_E) = -g(X_F,X_F)$. If this is $0$ at some point then $X_E$ and $X_F$ vanish at that point (since $g$ is positive definite on $E$ and negative definite on $F$) contradicting the non-vanishing of $X$. Hence $X_E$ and $X_F$ are non-vanishing sections of $E$ and $F$.

Conversely, if $X_E$ and $X_F$ are non-vanishing sections of $E$ and $F$ respectively, then up multiplying $X_F$ them by a positive function, we can assume that $g(X_E,X_E) = -g(X_F,X_F)$. Hence $X_E+X_F$ is a nowhere vanishing null vector field. CQFD

There are thus topological obstructions to the existence of such a vector field (mainly the non-vanishing of the Euler class of $E$ or $F$). For instance, Let $(A,g_A)$ and $(B,g_B)$ be Riemannian manifolds, with $A$ of non-zero Euler characteristic, and consider $(M,g) = (A\times B, g_A \oplus -g_B)$. Then $M$ does not admit a nowhere vanishing null vector field. Indeed, we have the splitting $TM = TA\oplus TB$, and the projection of a null vector field to TA must vanish somewhere since the Euler class of TA is non-zero.