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Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ would be one example, but are there other examples. Intuitively I would guess there are no other examples since going around loop on which a vector is parallel transported already leads to the fact that one ends up with a different vector than with which one started.

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  • $\begingroup$ Let $X(t)=x^{i}(t)(\frac{\partial}{\partial x^{i}})_{c(t)}$ where $X(t)$ is the tangent vector field defined on c(t) a parameterized curve on the manifold. Then the covariant derivative $\frac {DX}{dt}=0$. When say constant, constant relative to what? And by space-time do you mean a semi-Riemannian manifold? $\endgroup$ – Cinaed Simson Jul 26 '19 at 8:25
  • $\begingroup$ I mean a semi-Riemannian manifold and by constant I mean, that the covariant derivative vanishes along every path. Parallel transport along a path can be understood as holding a vector constant along that path. $\endgroup$ – yasalami Jul 26 '19 at 8:34
  • $\begingroup$ A weaker condition, that the symmetrization of the covariant derivative should vanish, is (equivalent to) the definition of a Killing vector field en.wikipedia.org/wiki/Killing_vector_field Killing vector fields may or may not exist, depending on the metric. $\endgroup$ – Robin Ekman Jul 26 '19 at 21:48
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An important concept to reason about these things is holonomy, which describes how the tangent space at a given point transforms by parallel transport along the closed path starting and ending at this point. For orientable Lorentzian manifold holonomy is an element of a Lorentz group $SO(1,3)$. All such elements, for all the paths comprise the holonomy group of a given manifold. For a “generic” curved spacetime holonomy group is the whole Lorentz group, but if the holonomy group is the proper subgroup we have a manifold of special holonomy. If the action of holonomy group leaves fixed a vector then there is a nontrivial parallel vector field which is precisely the object OP is interested in. Of course, since the full Lorentz group does not have invariant subspaces only special holonomy manifolds could admit parallel vector field.

Example. Consider the manifold that is a direct product of curved 3D Riemannian space and a timelike factor $(\mathbb{R},-dt^2)$ . The metric for such a spacetime could be written as: $$ ds^2 = - dt^2 + g^{(3)}_{ij}(X)\,dX^idX^j.$$ It is easy to see that the parallel transport of a vector $A =\alpha\, \partial_t $ ($\alpha=\mathrm{const}$) would leave it unchanged. So $A$ is a parallel vector field on this static spacetime.

For Riemannian manifold de Rham decomposition theorem is a useful tool in classifying such parallel vector fields and manifolds that admit them, but situation is more complicated in the Lorentzian case. There we have a class of spacetimes with parallel light-like vector field that has some importance in string/M theory. For a sampler, I suggest looking at this paper and follow the references.

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It is difficult to define the notion of a 'constant' vector field in curved manifold. This is because the vectors transform differently at different points in such a geometry. Moreover, Cartesian coordinates cannot be introduced globally. This gave rise to the concept of parallel transport. So if we have a vector field and we make a parallel transport, the vectors should change from point to point. However, the covariant derivative of a geodesic along its own tangent vector is always zero: $DV^\mu=0$, where $V^\mu$ is the tangent four-vector. In this sense, a geodesic is the natural extension of the definition of straight line to a curved manifold

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