In a paper the following definition is employed:
"Horizontal forms are forms on the total space that vanish when contracted against one or more vertical vector fields."
I am not that familiar with this notion. So my question: Does contraction of a form with a vector field mean something like taking the trace, i.e. $$tr(\omega^{i_1, i_2, \dots i_k} v_{i_j})$$? If so, to which indices does the statement refer? To any choice of an index?
Thanks for any help/advices!
No, in more simple terms it is literally "plugging in a (vertical) vector field into the differential form". Given a $k$-form $\omega$ and a vector field $X$, you can define a $(k-1)$-form by $$ \iota_x\omega(X_1, \dots,X_{k-1}):=\omega(X,X_1,\dots,X_{k-1}). $$ For horizontal forms $\alpha$, that means if $X$ is any vertical field, we have $$ \iota_X\alpha(X_1, \dots,X_{k-1})=\alpha(X,X_1,\dots,X_{k-1})=0. $$ The Wikipedia entry also does a good job of providing more details as well as any more graduate oriented textbook on differential geometry.