Timeline for Homogeneous polynomial vector fields tangent to the unit sphere

7 events

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Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Jan 30, 2015 at 10:06	vote	accept	Denis Serre
Jan 30, 2015 at 9:54	answer	added	Robert Bryant		timeline score: 4
Jan 30, 2015 at 8:47	comment	added	Joonas Ilmavirta		A continuation to my previous comment: If we assume $a_i^{j_1,\dots,j_d}$ to be symmetric in the upper indices, we lose redundancy and have a parametrization of all $d$-homogeneous $n$-vector fields. We can then write $a_i^{j_1,\dots,j_d}$ as a sum of two parts, one of which is symmetric under swapping the lower index with an upper one and the other one antisymmetric. Now $X\cdot v(X)=0$ iff the symmetric part vanishes. The problem is then to find the dimension of the space of tensors with these symmetries (assuming my reasoning makes sense).
Jan 30, 2015 at 8:36	history	edited	Denis Serre	CC BY-SA 3.0	added 63 characters in body
Jan 30, 2015 at 8:35	comment	added	Joonas Ilmavirta		You also have $D(n;1)=\frac12n(n-1)$ because a linear vector field $v(X)=AX$ satisfies $X\cdot v(X)=0$ iff the matrix $A$ is skew-symmetric. I wonder if it would be useful in general to write $v_i(X)=\sum_{j_1,\dots,j_d}a_i^{j_1,\dots,j_d}X_{j_1}\cdots X_{j_d}$ and study the symmetries of $a_i^{j_1,\dots,j_d}$.
Jan 30, 2015 at 8:13	history	asked	Denis Serre	CC BY-SA 3.0