Connection, Curvature, and Holonomy for Ehresmann Connections on Riemannian Submersions

We study Ehresmann connections on smooth fibered manifolds equipped with Riemannian submersion structures\[\pi : M \rightarrow Z,\]focusing on the interaction between horizontal transport and the induced metric geometry on the base manifold. Assuming Riemannian metrics on both the total space and the base, we analyze conditions under which parallel transport along the horizontal distribution acts isometrically and is compatible with the Riemannian submersion structure determined by $\pi$.Within this setting, metric-preserving horizontal transport induces local geometric consistency on $Z$ via the horizontal isometry $d\pi|_{H}$, identifying the horizontal distribution as the mediator between the intrinsic geometry of $(M,g_M)$ and the quotient geometry of $(Z,g_Z)$.We then examine the global structure of horizontal transport through curvature and holonomy invariants of the Ehresmann connection. By the Ambrose--Singer theorem, holonomy is characterized as the global obstruction to path-independent horizontal transport, while curvature provides its infinitesimal generator.In particular, vanishing horizontal curvature yields globally consistent metric transport on the base, whereas nontrivial curvature induces path-dependent identifications governed by holonomy. The O’Neill tensorial decomposition further clarifies how horizontal and vertical geometries interact and how this interaction affects compatibility with the Riemannian submersion structure.The resulting framework unifies Ehresmann connection theory, Riemannian submersion geometry, and holonomy theory within global differential geometry, isolating the precise relationship between local metric compatibility and global geometric obstructions in fibered Riemannian manifolds.