(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A linear vector field on R^n, defined by a matrix A, is a Killing field if and only if A is antisymmetric.

2. Relevant equations

3. The attempt at a solution

I took R^2. Let A be the following matrix:

0 -10

10 0

A is antisymmetric. However, how in the heck is A a Killing field? For example, let

v = (1, 10), then Av = (-100, 10)

u = (2, 3), then Au = (-30, 20)

and <u, v> = 32 while <Au, Av> = 3200, so the dot product after we apply A is a multiple of the original dot product by the determinant of A (in this case it's 100).

What exactly am I misreading? The only restriction on A is that it is antisymmetric. A Killing field is a vector field that preserves the metric, i.e. in particular it will preserve the dot products of vectors.

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# Homework Help: Killing Fields on R^n

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