In the question given above, H is the mean curvature of the immersed real projective space in the Euclidean m-space and c2n denotes the volume of the unit 2n-sphere.
If the immersion is not isometric, then the pullback f^*\delta of the Euclidean metric \delta does not coincide with the given metric g on the Riemannian real projective space (RP^2n, g).
The pullback does induce a Riemannian structure and by Chen's theorem mentioned above, the inequality is satisfied for this metric. Expressing the original inequality in terms of this one for the induced structure, one is tasked with bounding the difference of the induced volume measure against the given volume measure.
As far as I can tell, this is almost arbitrarily bad. I believe that the inequality holds with > 0 on the RHS, but it is hard for me to imagine a uniform bound (as in Chen's theorem) unless we have more information on the difference between the metric induced by the immersion and the given metric.
Dear Professor Wheeler:
Thank you very much for your comment/answer to my question.
I shall mention that in my article published in [Bull. Inst. Math. Acad. Sinica 7 (1979), no. 3, 301–311] I have assumed that the metric on the real projective space is the standard one (I shall mention it in my earlier remark).
Bang-Yen
- Badges
- Science topic
- Similar topics
- Mathematics