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Riemannian metric question

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Poirot

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Feb 15, 2012
250
I'm interested in part iv) on the attachment. This is my work so far:
e=(1,0) and e'=(0,1) form a basis of the tangent space at any point z=(x,y). Making the identification (x,y)->x+iy, we get g(e,e')=0 and g(e,e)=g(e',e')=$\frac{1}{im(z)^2}$.

a(t)=z+t and b(t)=z+it are generating curves for e,e' respectively.
(lets call the function f)

$f(z)=\frac{z}{|z|^2}$ so $f(a(t))=\frac{z+t}{|z+t|^2}$. I need to find f'(a(t)) to proceed. How can I cope with differentiating the modulus of a complex number z? Thanks
 

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ZaidAlyafey

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MHB Math Helper
Jan 17, 2013
1,667
Re: riemannian metric question

I don't know about metric spaces , but I know about complex analysis ... To differentiate a function a necessary requirement is to satisfy the cauchy-riemann equation .. suppose that \(\displaystyle f(z)=|z|\) this function is clearly not differentiable

\(\displaystyle f(z)=\sqrt{x^2+y^2} \)

By the cauchy-reimann equation we must have \(\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\)

Which is clearly not satisfied for \(\displaystyle |z|\)

The function you are trying to differentiate seems a function of several variables ? , are you differentiating with respect to t ?
 
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Poirot

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Feb 15, 2012
250
Re: riemannian metric question

I need to find the differential of f at z evaluated at e (and e'). This is equal to f'(a(t)) evaluated at t=0.