In "The Large Scale Structure of Space and Time" by Hawking and Ellis the following is stated:
"There will be various fields on $M$, such as the electromagnetic field, the neutrino field, etc., which describe the matter content of space-time"
Here $M$ is meant to be a spacetime (i.e. a smooth $4$-dimensional manifold). From a mathematical perspective, how does one interpret "field" as quoted above. Are the authors saying that these matter fields (such as the electromagnetic field and the neutrino field) are vector fields on $M$?