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Jeffrey Lee in his book "Manifolds and differential geometry" defines the notion of Ehresmann connection as

Definition 12.12. A (linear Ehresmann) connection on a vector bundle $\pi: E \to M$ is a smooth distribution $\mathcal{H}$ on the total space $E$ such that

  • $\mathcal{H}$ is complementary to the vertical bundle: $TE = \mathcal{H} \oplus \mathcal{V} \ E$
  • $\mathcal{H}$ is homogenous: $T_y \mu_r (\mathcal{H}_y) = \mathcal{H}_{ry}$, where $\mu_r: E \to E $ is the multiplication map.

He also gives it as an exercise to prove that there is a bijection between such connections and covariant derivatives (which he calls Koszul connections).

However M. Postnikov in his book "Semester IV. Differential Geometry" gives a definition using local-coordinate representation of $\mathcal{H}$, such that the last condition above is replaced by:

  • the condition that in the trivialization $E|_U \cong U \times V$, with $V$ - vector space, and $x_1,...,x_n$ - coordinates on the base, $a_1,...,a_m$ - coordinates on the fiber, the forms $\theta_i$ defining $\mathcal{H}=Ann(\{\theta_i\})$, have the form $$\theta_i=da^i + \Gamma_{kj}^{i}(x) a^j dx^k \ \ (*)$$or in other words that if $$\theta_i=da^i + e_k^i(x,a)dx^k \ \ (**)$$ then functions $e_k^i(x,a)$ must be linear in the coordinates on the fiber.

Now, it is easy to see that Postnikov's definition satisfies all the conditions of Lee's definition.

However if one tries to write coordinate representation of Lee's definition, one can get to the form $(**)$ with functions $e_k^i(x,a)$ being only homogenous in the coordinates on the fiber.

M. Postnikov then uses his second condition in the explicit construction showing the equivalence of his notion of connection to the usual covariant derivate - basically, $\Gamma_{kj}^{i}(x) dx^k$ give connection forms $\omega_j^i$ on the base. He also later emphasizes, that, contrary to the case of principle bundles, one cannot make an easy coordinate free definition in the case of vector bundles. From that I suspect that it might be actually impossible to retrieve a covariant derivative out of Lee's definition - there seems to be no way of separating coordinates on the base from coordinates on the fiber in $e_k^i(x,a)dx^k$ to get connection forms on the base.

So can you please clarify what is the right take on the notion of Ehresmann connection? Is there a clean coordinate-free way of defining it similar to Lee's approach, contrary to what M. Postnikov is suggesting?

Remark. I am familiar with an approach of passing to the frame bundle and defining there a connection using an equivariant fundamental form. I am curious if it is possible to work only on the vector bundle itself.

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The fact you're missing is that all differentiable homogeneous functions (of degree 1) are linear! Lee gives this as Lemma 12.26, and the proof is essentially one line: if $f : V \to W$ is a homogeneous map between real vector spaces, compute $$Df|_0(v) = \frac d{dt}\Big|_{t=0}f(tv)=\frac d{dt}\Big|_{t=0}tf(v)=f(v),$$ so $f$ is the linear map $Df|_0.$

Since you are assuming $\mathcal H$ is a smooth distribution, you should find that the functions $e^i_k$ are smooth and homogeneous in the fibre, so applying this lemma you can conclude that they are in fact linear in the fibre.

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  • $\begingroup$ Oh, wow,thanks! I must have overlooked it! $\endgroup$
    – Bananeen
    Feb 21 '18 at 3:36

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