Let $(M,F)$ be a Riemannian foliation, i.e. there is a metric of $TM$ such that $g$ is bundle-like (locally $g_Q=g_{ij}(y)\,dy^i\otimes dy^j$ for a foliated chart $(x,y)$ and $Q=TM/F\cong F^\perp$).
A principal bundle $P\to(M,F)$ is called foliated, if it is quipped with a lifted foliation $F_P$ invariant under the structure group action, transversal to the tangent space to the fiber and $TF_P$ projects isomorphically onto $TF$.
For an associated foliated vector bundle $E$, an adapted connection is called basic, if the connection $1$-form $\omega$ on its principal bundle $P^E$ satisfies $$L_X\omega=0,$$ for any $X\in\Gamma(F_p)$.
I have a question about the existence of the basic connection.
Q
Let $E$ be foliated transversal Clifford bundle that is a olaited bundle of $Cl(Q)$-Clifford modules with compatible connection $\nabla^E$. Why it is always possible to choose a basic connection for $E$? (from the book, Masayuki Asaoka Aziz El Kacimi Alaoui Steven Hurder Ken Richardson , Foliations: Dynamics, Geometry and Topology, Page 168).
From the paper,J. Brung, F. W. Kamber, and K. Richardson , INDEX THEORY FOR BASIC DIRAC OPERATORS ON RIEMANNIAN FOLIATIONS, authors wrote that: basic connections always exist on a foliated principal bundle over a Riemannian foliation. from the book Foliated Bundles and Characteristic Classes. Why it is true?
PS: I think the first problem is from the property that the group of autormphism preserving the foliation is compact. For the second problem, I guess the authors mean that the Lie group foliated principal bundle is compact?
