# Inducing linear connections via functors

Let $$M$$ be a smooth manifold and let $$\pi:E\rightarrow M$$ be a real vector bundle over it. Let $$\nabla$$ be a linear (Koszul) connection on $$E$$ (here in this question I am using covariant derivatives, but if it is more convenient, answers may treat linear connections as horizontal distributions on $$E$$).

It is a well known fact that then there are naturally induced linear connections on "related" vector bundles, such as $$E^\ast,\ \bigotimes^kE$$, etc.

There is of course a systematic way of describing these induced connections from the point of view of principal fiber bundles, however that approach is only of marginal interest for the purposes of this question.

If $$\mathbb A$$ is an indexing set, $$(U_\alpha)_{\alpha\in\mathbb A}$$ is an open cover of $$M$$, and $$(\varphi_{\alpha\beta})$$ is a cocycle of transition functions (with $$\varphi_{\alpha\beta}:U_{\alpha\beta}=U_\alpha\cap U_\beta\rightarrow\text{GL}(V)$$, $$V$$ is a finite dim. real vector space), then a unique vector bundle may be constructed from these data, which is denoted as $$\text{Vb}(\varphi_{\alpha\beta})$$. This construction is nicely outlined in for example Natural Operations in Differential Geometry by Michor, Kolár, Slovák.

The general way of constructing "related" vector bundles is also stated here. Let $$\text{Vec}_\mathbb R$$ denote the category of finite dimensional real vector spaces, and let $$\mathcal F:\text{Vec}_\mathbb R\rightarrow\text{Vec}_\mathbb R$$ be a functor from this category to itself. If for any $$A\in\text{Hom}(V,W)$$, the assignment $$A\mapsto\mathcal F(A)\in\text{Hom}(\mathcal F(V),\mathcal F(W))$$ (or $$\text{Hom}(\mathcal F(W),\mathcal F(V))$$ in case of a contravariant functor) is a smooth map, then we call $$\mathcal F$$ a smooth functor.

Now, if $$\mathcal F$$ is a smooth covariant functor, then we define a new vector bundle $$\mathcal F(E)$$ by $$\text{Vb}(\mathcal F(\varphi_{\alpha\beta}))$$ and if it is a contravariant functor then by $$\text{Vb}(\mathcal F(\varphi_{\alpha\beta}^{-1}))$$. This is extended in an obvious way to multiple vector bundles and multifunctors.

However from this it seems natural to me that given

• a vector bundle $$E\rightarrow M$$;

• a linear connection $$\nabla\equiv\nabla(E)$$;

• a smooth functor (covariant or contravariant) $$\mathcal F$$

there should be an induced linear connection $$\nabla(\mathcal F(E))$$ on $$\mathcal F(E)$$ by this functor.

Question 1: Assuming this is true, how does this functor exactly induce this connection on the vector bundle $$\mathcal F(E)$$?

Question 2: How is this process related to the usual way of inducing connections on vector bundles associated to principal bundles? Considering the fact that for functorial vector space operations like taking duals, direct sums, tensor products etc. there are corresponding operations on representations, I assume if we are given a principal bundle $$P\rightarrow M$$ with structure group $$G$$, and a representation $$\rho:G\rightarrow\text{GL}(V)$$, then there is a corresponding representation $$\mathcal F(\rho)=\mathcal F\circ \rho$$ ($$\mathcal F$$ is covariant) or $$\mathcal F(\rho)=\mathcal F\circ\rho^{-1}$$ ($$\mathcal F$$ is contravariant), and the induced connections for associated vector bundles induced by these representations are the connections I am looking for, but I would be happy if I could get some links to some papers or textbooks that discuss this approach in greater detail.

• I can't give now a complete formalization, but I believe that one should consider not arbitrary functors F, but something like Schur functors. Those one can functorially lift to situations where a group acts, or a Lie algebra (such as the Lie algebra of vector fields). Probably one can formalize then in what sense this holds also for linear connections. – Sasha Feb 2 at 21:20

Let $$(U_i)$$ be a covering, locally, the Koszul derivative is defined by a $$1$$-form $$\omega_i:U_i\rightarrow gl(V)$$ such that for every section $$Y$$, $$\nabla_XY=dY(X)+\omega_i(X)Y$$, since $${\cal F}$$ is smooth, one can define $${\cal F}(\omega_i)$$ by $${\cal F}(\omega_i)(X)$$. This makes sense since $$\omega_i(X(x))\in gl(V)$$ thus is a morphism of $$V$$.
For a section $$Y$$ of $${\cal F}(E)$$ on $$U_i$$, one can set $$\nabla_XY=dY.X+{\cal F}(\omega_i)(X)(Y)$$.
• I have thought about this, but my issue is that it seems to me if $\mathcal F$ is contravariant, then the connection form should get a negative sign. Eg. if it is the dual space funtor, then we have $\nabla(E^\ast)=d-\omega^\ast$. On the level of the Lie group it seems natural to invert if a contravariant functor is involved (f. ex. my definition of related vector bundles induced by a contravariant functor), and of course I know that inverse on the group level is additive inverse on the Lie algebra level, but nontheless... – Bence Racskó Feb 3 at 9:05