Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hi,

Can one define a Fubini-Study metric/Kaehler metric on the projective space of an infinite dimensional Hilbert space, i.e. using the formula $\partial \bar{\partial} \log |Z|^2$? This should be very well-known to the experts. Anyhow I don't have much experience with infinite dimension and worried that something may go wrong. I appreciate any comments or references.

share|improve this question
add comment

2 Answers

up vote 3 down vote accepted

Maybe it's easier to see that the definition extends if you don't use a formula:

The unit sphere inherits a Riemannian metric from the Hilbert space in the standard manner and since it is invariant under the circular symmetry $(e^{i\theta},x) \mapsto e^{i\theta} x$, it will project down to a Riemannian metric on the complex projective space.

share|improve this answer
    
Thanks Ahmed and Alvarez, so definition works as in the finite dimensional case. –  Kiu Apr 2 '13 at 3:20
add comment

The answer is yes and can be found for example in S.Kobayashi "The geometry of bounded domains" T.A.M.S 1959 (92) 267- 290

share|improve this answer
    
Just a small correction. I assummed taht you mean separable Hilbert space. For wild Hilbert spaces not allowing countable orthonormal basis the above answer does not apply –  Ahmed Sulejmani Apr 1 '13 at 13:46
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.