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I apologize for the long question. My ultimate aim is to understand the existing terminology and find appropriate terminology for covariant derivatives on vector bundles which generalizes to "covariant" derivatives on fiber bundles (a linear Ehresmann connection is to a [linear] covariant derivative as a nonlinear Ehresmann connection is to [fill in the blank]). Or if such terminology already exists, to find out what it is.

A covariant derivative on a vector/tensor bundle $E \to M$ is an $\mathbb{R}$-linear map of the form $\nabla \colon \Gamma(E) \to \Gamma(E \otimes T^*M)$. As I understand it, the "covariant" part of this comes from the fact that the $T^*M$ component changes covariantly under coordinate changes and not how the $E$ component changes. Is this correct? A pedantic followup question is prompted by some inconsistency I've found in various literature: Must $E$ be a tensor bundle induced from tangent/cotangent vector bundles for the "covariant" qualifier to still apply?

The motivation for the qualifier "covariant" seems to ultimately stem from coordinate-based definitions and considerations. I've also seen uses of "covariant" to mean "independent of coordinate choice" in a more abstract setting. Is this also valid? Does the notion and terminology of a Koszul connection supplant the coordinate-minded covariant derivative? I'm guessing that the $\nabla$ formulation as a map producing a section of a tensor bundle came from Koszul.

Linear Ehresmann connections are in one-to-one correspondence with covariant derivatives/Koszul connections, and there is a notion of a nonlinear Ehresmann connection on a fiber bundle. I have come up with a corresponding definition for a "nonlinear covariant derivative/Koszul connection" on a fiber bundle, which has as a natural example in the fiber bundle $\Pr_2^{S\times M} \colon S \times M \to M$, where $M$ and $S$ are smooth manifolds (noting that $\Gamma(\Pr_2^{S\times M})$ can be identified naturally with $C^\infty(M,S)$). Under certain natural identifications, if $\nabla$ denotes this nonlinear operator (definition left out to avoid clutter) and $\phi \in C^\infty(M,S)$, then $\nabla \phi = T \phi$, i.e. this connection/derivative is the tangent map operator. This suggests the question: Can the tangent map operator be considered a covariant derivative? It is certainly independent of any coordinate choice.

Apart from the questions interspersed throughout the text above, my main question is: Is there already terminology for this "nonlinear covariant derivative". If not, would "nonlinear covariant derivative" or "nonlinear Koszul connection" be appropriate for such? The proliferation of "connections" (e.g. Levi-Civita, Cartan, Ehresmann, Koszul, affine, etc.) suggests that "connection" is preferred over "derivative", is this correct?

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As I understand it, the "covariant" part of this comes from the fact that the T∗M component changes covariantly under coordinate changes and not how the E component changes. Is this correct?

Yes.

The motivation for the qualifier "covariant" seems to ultimately stem from coordinate-based definitions and considerations. I've also seen uses of "covariant" to mean "independent of coordinate choice" in a more abstract setting. Is this also valid?

Yes on this one as well.

Is there already terminology for this "nonlinear covariant derivative". If not, would "nonlinear covariant derivative" or "nonlinear Koszul connection" be appropriate for such?

There is a notion of nonlinear connection, even for the tangent bundle. Recall that the auto-parallel curves of the Levi-Civita connection of a metric are the geodesic. Recall also that these curves have a variational characterization, the local minimizers of length. The geodesic equations are the Euler-Lagrange equations of the length functional. The metric is used to define the lagrangian of this functional. More generally, given a lagrangian on a manifold, i.e., a function on the total space of the tangent bundle, the extremals of the resulting functional can be interpreted as the auto-parallel curves of a nonlinear connection on the tangent bundle. It is called nonlinear because the parallel transport defined by this connection is a nonlinear map. Finsler geometry is a special incarnation of this philosophy. The lagrangean in this case is a function whose restriction to each tangent space is a norm on that tangent space.

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Hmm.. I'll see if my definition is compatible with that of Finsler geometry. Thank you. –  Victor Dods Jan 11 '12 at 3:16

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