How the inner product changes under non-linear transformation

[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 ?

 

[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.

 

[mnb96]

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