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In Dieudonne's Treatise on Analysis (Volume III Section 17.16) the following definition for a linear connection in a vector bundle is given:

A linear connection $C$, in a vector bundle $(E,\pi, M)$ is defined to be a smooth map $C:TM\oplus E\to TE$ which satisfies for each $(k_x,u_x)\in T_xM\oplus E_x$, the following conditions:

  1. $\pi_E(C(k_x,u_x))=u_x$, where $\pi_E:TE\to E$ is the standard projection
  2. $T\pi (C(k_x,u_x))=k_x$ where $T\pi:TE\to TM$ is the tangent mapping
  3. The mapping $k_x\mapsto C(k_x,u_x)$ is a linear mapping of $T_xM$ into $T_{u_x}E$
  4. The mapping $u_x\mapsto C(k_x,u_x)$ is a linear mapping of $E_x$ into $(TE)_{k_x}$, which is the fiber of $T\pi:TE\to TM$ lying over the base point $k_x\in TM$.

Geometrically this definition makes sense; I interpret $C(k_x,u_x)$ as "the direction I should move within $E$ in order to reach an infinitesimally close fiber, provided that I start at the point $u_x$ of the fiber $E_x$, and move in the direction $k_x$ in the base manifold" (picture below for one-dimensional base and fibers).

With the usual abuse of notation, we have local coordinates $(x,\dot{x},u,\dot{u})\in \Bbb{R}^n\times \Bbb{R}^n\times\Bbb{R}^p\times\Bbb{R}^p$ on $TE$, where $x$ describes the base point, $\dot{x}$ describes the vectors in the fiber tangent to $M$, $u$ describes points in the fibers of $E$ and $\dot{u}$ describes the points tangent to $E$. Relative to such adapted coordinates, the mapping $C$ takes the form $C: (x,\dot{x},u)\mapsto (x,\dot{x},u,-\Gamma_x(\dot{x},u))$, where $\Gamma_x:\Bbb{R}^n\times\Bbb{R}^p\to\Bbb{R}^p$ is bilinear (the minus sign is merely convention to make things match up with the usual Christoffel symbols).

Question.

Suppose we are given two vector bundles $E,F$ over $M$, with respective linear connections $C_E,C_F$. How can I define abstractly (i.e without any references to coordinates) a connection $C_{\text{Hom}(E,F)}$ in the bundle $\text{Hom}(E,F)$? I have already managed to define it in coordinates, meaning I figured out how, given local coordinates, to define the $\Gamma$ for $\text{Hom}(E,F)$ in terms of $\Gamma$ for $E$ and $F$. Then, I used the appropriate transformation laws for $\Gamma$'s to check that everything works out so that we do indeed obtain a globally well-defined mapping $\text{C}_{\text{Hom}(E,F)}$.

I will include this local definition if anyone requests it; but the notation is a little heavy so I suppress it for now. The reason I ask this question is because defining things locally and then verifying the compatibility conditions is a little tedious, especially since $\Gamma$ has a somewhat complicated transformation behavior.

By the way, I have already determined how to define connections on $E\oplus F$ and $E\otimes F$, in an abstract manner (and also locally). Also, I'm aware that $\text{Hom}(E,F)\cong E^*\otimes F$, so if someone can provide an abstract definition for $C_{E^*}$ in terms of $C_E$ that is also sufficient. Thank you.


Picture:

Connections

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This is by no means a comprehensive answer, but it outlines a potentially useful intrinsic description of the compatibility condition of various connections. I haven't done all of the computations to show that everything is well-defined and smooth, and doing so will probably require some amount of coordinates.

Every smooth vector bundle $(E,\pi,M)$ has a secondary vector bundle structure $(TE,\pi_*,TM)$. This is needed for condition 4 to have an intrinsic meaning, since it requires a vector space structure on the fibers of $\pi_*$. The vector bundle operations can be defined as follows: given $a_{u_x},b_{v_x}\in TE$ with $\pi_*(a_{u_x})=\pi_*(b_{u_x})=w_x$, choose a curve $\gamma:(-\epsilon,\epsilon)\to M$ with $\gamma'(0)=w_x$, lift $\gamma$ to two curves $\alpha,\beta:(-\epsilon,\epsilon)\to E$ with $\alpha'(0)=a_{u_x}$, and $\beta'(0)=b_{v_x}$, and define $c_1a_{u_x}+c_2b_{v_x}:=(c_1\alpha+c_2\beta)'(0)$. This map will not depend on the choice of curves.

Let $(E^*,p,M)$ be the dual vector bundle and $\delta:E^*\times_ME\to\mathbb{R}$ denote the natrual pairing. By differentiating along lifts of curves as before, we can obtain a map $\delta_*:TE^*\times_{TM}TE\to\mathbb{R}$ which is bilinear w.r.t. the secondary vector bundle structures on $TE^*$ and $TE$. The dual connection on $E^*$ is then the unique connection $C_{E^*}:TM\times_ME^*\to TE^*$ satisfying $$ \delta_*(C_{E^*}(v_x,\mu_x),C_E(v_x,u_x))=0 $$ for all $x\in M,v_x\in T_xM,\mu_x\in E^*_x,u_x\in E_x$. In essense, this is saying that the dual connection is uniquely determined by the requirement that the pairing of parallel sections along curves is constant. This is equivalent to the more common condition $\nabla_v\langle\mu,U\rangle=\langle\nabla_v\mu,v\rangle+\langle\mu,\nabla_vU\rangle$, since linear connections are determined by their parallel sections along curves.

One can repeat this process to obtain compatibility conditions from other liberwise (multi)linear morphisms, such as the Whitney sum $E\leftarrow E\oplus F\to F$, the tensor product $E\times_M F\to E\otimes F$, or the evaluation map $\operatorname{Hom}_M(E,F)\times_ME\to F$.

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  • $\begingroup$ Thank you for this answer, you've pinpointed the appropriate compatibility condition I was looking for, and the sentence "In essence, this is saying that the dual connection is uniquely determined by the requirement that the pairing of parallel sections along curves is constant" made it click for me. I will verify the more technical details a day or two later, but that shouldn't be too hard. btw in the last paragraph, you say "... fiberwise linear morphisms", but surely you mean bilinear right, because the tensor product mapping and evaluations are bilinear. $\endgroup$
    – peek-a-boo
    Jun 12 '21 at 14:56
  • $\begingroup$ btw as an aside, do you know of the name given to this type of connection, or if any other source defines it in this way? because when I search for connections, all I see is the definition involving $\nabla$ (the very first one I learnt, but I always had reservations about it since it feels too algebraic for my taste) and I see more abstract stuff involving principal bundles and associated vector bundles $\endgroup$
    – peek-a-boo
    Jun 12 '21 at 15:03
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    $\begingroup$ @peek-a-boo "fiberwise multilinear" would probably be most precise. I haven't seen this specific definition before, but it's equivalent to a linear Ehresmann connection. These are normally defined by a horizontal distribution $HE\subseteq TE$ which is complimentary to the vertical distribution $VE:=\ker(\pi_*)$. This allows one to uniquely lift tangent vectors from $T_xM$ to $H_{v_x}E\subset T_{v_x}E$ via the horizontal lift $\operatorname{hl}_{v_x}:T_xM\to T_{v_x}E$. $C$ is exactly this lifting map. $\endgroup$
    – Kajelad
    Jun 12 '21 at 20:00
  • $\begingroup$ It'll probably take me a day or two to digest all the details, but they seem pretty routine. Thanks once again for your help! $\endgroup$
    – peek-a-boo
    Jun 12 '21 at 21:26

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