If X and Y are both these constructs, then the Lie bracket of X and Y is also one of these things, which can be defined as sections of tangent bundles. Every one of these constructs on a compact manifold is complete. Varieties of these constructs that preserve their metric on a Riemannian manifold are named for (*) Wilhelm Killing. The Kelvin-Stokes theorem relates a line integral over one of these constructs to the surface integral of its curl. 
If X and Y are both these constructs, then the Lie bracket of X and Y is also one of these things, which can be defined as sections of tangent bundles. Every one of these constructs on a compact manifold is complete. Varieties of these constructs that preserve their metric on a Riemannian manifold are named for (*) Wilhelm Killing. The Kelvin-Stokes theorem relates a line integral over one of these constructs to the surface integral of its curl. For 10 points, identify these constructs which assign a direction and magnitude to each point in a space
If X and Y are both these constructs, then the Lie bracket of X and Y is also one of these things, which can be defined as sections of tangent bundles.
vector fields [prompt on fields; accept Killing fields until "Killing" is read and anti-prompt after]