Vector-valued Differential Form - Basic or Tensorial Forms On Principal Bundles

Basic or Tensorial Forms On Principal Bundles

Let E → M be a smooth vector bundle of rank k over M and let π : F(E) → M be the (associated) frame bundle of E, which is a principal GL_k(R) bundle over M. The pullback of E by π is isomorphic to the trivial bundle F(E) × Rk. Therefore, the pullback by π of an E-valued form on M determines an Rk-valued form on F(E). It is not hard to check that this pulled back form is right-equivariant with respect to the natural action of GL_k(R) on F(E) × Rk and vanishes on vertical vectors (tangent vectors to F(E) which lie in the kernel of dπ). Such vector-valued forms on F(E) are important enough to warrant special terminology: they are called basic or tensorial forms on F(E).

Let π : P → M be a (smooth) principal G-bundle and let V be a fixed vector space together with a representation ρ : G → GL(V). A basic or tensorial form on P of type ρ is a V-valued form ω on P which is equivariant and horizontal in the sense that

1. for all g ∈ G, and
2. whenever at least one of the v_i are vertical (i.e., dπ(v_i) = 0).

Here R_g denotes the right action of G on P for some g ∈ G. Note that for 0-forms the second condition is vacuously true.

Given P and ρ as above one can construct the associated vector bundle E = P ×_ρ V. Tensorial forms on P are in one-to-one correspondence with E-valued forms on M. As in the case of the principal bundle F(E) above, E-valued forms on M pull back to V-valued forms on P. These are precisely the basic or tensorial forms on P of type ρ. Conversely given any tensorial form on P of type ρ one can construct the associated E-valued form on M in a straightforward manner.