# Examples of connection preserving maps in differential geometry

In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other algebraic gadget, it seems natural to consider morphisms $$f:M \to N$$ that preserve a chosen linear connection $$(M, \nabla), (N, \nabla')$$. I've tried searching through the differential geometry literature, and connection-preserving morphisms doesn't seem to be discussed very much (except the case where an isometry between Riemannian manifolds induces a connection-preserving morphism between their Levi-Civita connections).

Have connection-preserving maps been considered by differential geometers, and if so, can someone point me in that direction?

• I've been trying to get the term 'connectomorphism' going for some time. May 21 '20 at 19:45
• For a prequantum bundle of a symplectic manifold, the group of connection-preserving automorphisms is known as the quantomorphism group ncatlab.org/nlab/show/quantomorphism+group. In the context of general principal bundles, the group of gauge transformations that preserve a given connection is finite dimensional and isomorphic to the centralizer of the holonomy group. This observation can be used to classify all possible groups that arise as connection-preserving gauge transformations. See eg arxiv.org/abs/hep-th/0203027 May 22 '20 at 18:42
• In chapter 6 of Kobayashi-Nomizu "Foundations of Differential Geometry vol. 1" these are covered under the keyword "affine mapping" May 24 '20 at 21:27
• Affine geometry is maybe a topic of interest, see projecteuclid.org/euclid.aspm/1543086326 May 25 '20 at 14:32