TITLE: Why is Gauss credited with this connection?
QUESTION [11 upvotes]: Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi_* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose fiber over a point $b$ is the cohomology group $H^k(X_b, \mathbb Z)$.
We can tensor this system by $\mathcal O_B$ and obtain a holomorphic vector bundle $E^k \to B$. This bundle is equipped with a flat connection $\nabla$, that is induced by the exterior derivative $d$ in local coordinates. This connection is called the Gauss-Manin connection of the bundle $E^k$.
Now: Why do we call this the Gauss-Manin connection, when it seems that nothing that Gauss could have worked on relates to it?

REPLY [18 votes]: I've heard from a reliable source that Grothendieck decided on this name because he wanted to include the
first person who discovered the thing and the last person who made an important contribution:-)
(Of course, Grothendieck knew that there were many in between).
However, as it frequently happens, this attribution of the first discovery is not correct.
The fact that periods of an elliptic integral satisfy a hypergeometric equation was discovered
by Legendre. Actually several important discoveries of Legendre are attributed to Gauss,
(for example, quadratic reciprocity law, and the method of least squares)
and Legendre bitterly complained about this himself. 
Also the discovery of the arithmetic geometric mean is universally credited to Gauss, while Lagrange discovered it and published 6 years earlier, and Legendre used it to compute elliptic integrals.