# higher order structure by higher order derivatives

Anyone recall a structure determined by a 3rd order partial derivative? not the general nth order of recent Baranovsky

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I don't understand the question. What does "structure determined by a 3rd order partial" mean? –  Hans Stricker Jul 23 '11 at 14:49
Have you considered having a look at math.stackexchange.com/questions/14841 ? Or else, to repeat the question of Hans: what is your question? –  András Bátkai Jul 23 '11 at 19:33
Just in case (but I am to far from this topic): a foliated $3$-web in the plane is linearizable if and only if its curvature is $\equiv0$. Isn't true that this curvature involves the third derivatives of the vector field defining the foliations ? –  Denis Serre Jul 23 '11 at 19:56

## 2 Answers

The Schwarzian derivative is third-order and plays an important role in the geometry of the projective line.

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The associativity condition for the symmetric 3-tensor in a Frobenius manifold is a third-order PDE on the potential: the so-called WDVV equation.

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Jos\'e, That is the one I was after. Apologies for the vagueness of my question,thinking it did have something to do with Frobenius manifolds, but having not found the answer in Dubrovin, I thought that too might be a faulty memory. –  Jim Stasheff Jul 25 '11 at 11:48
Jim, glad to have been of help! –  José Figueroa-O'Farrill Jul 25 '11 at 12:00