Lifts of maps to frame bundles

Let $(M,g)$ be a Riemannian manifold, $L(M)$ be its frame bundle, $O(M)$ its orthonormal frame bundle. For a distribution $D$ on $M$ we define a subbundle $L(D)\subset L(M)$ or $O(D)\subset O(M)$ in a natural way. This allows us to consider a lift $L\phi$ of a map $\phi :M\to N$ not necessarily being a local diffeomorphism. More precisely, if $\phi :M\to N$ is a submersion, then $L\phi :L({\mathcal{H}}^{\phi })\to L(N)$ or $L\phi :O({\mathcal{H}}^{\phi })\to L(N)$, where ${\mathcal{H}}^{\phi }$ is a horizontal distribution of $\phi$. Equipping $L(M)$ and $L(N)$ with the Mok metrics, we study conformality and harmonicity of lifts $L\phi$.