How the inner product changes under non-linear transformation

1. Nov 24, 2012

mnb96

Hi,

if we suppose x and y are two elements of some vector space V (say ℝn), and if we consider a linear function f:V→V', we know that the inner product of the transformed vectors is given by: $$\left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} , \overline{f}f\mathbf{y} \right\rangle = \left\langle \overline{f}f\mathbf{x} , \mathbf{y} \right\rangle$$ where $\overline{f}$ is the adjoint operator of $f$.

What can we say about $\left\langle f\mathbf{x} , f\mathbf{y} \right\rangle$ when f is non-linear, for example a diffeomorphism ?

Last edited: Nov 24, 2012
2. Nov 25, 2012

Muphrid

I admit I'm not well versed in stuff with manifolds and such, but isn't a diffeomorphism basically a way of mapping positions on one manifold to positions on another? If so, then $f$ is just the Jacobian of the mapping, and it is inherently dependent on position within the manifold. It's important to note that positions won't obey this transformation law, only the full, nonlinear transformation.

3. Nov 27, 2012

mnb96

Hi Muphrid,
thanks for the answer. That's exactly what I wanted to know.