Timeline for Existence of complementary pairs of foliations on spheres
Current License: CC BY-SA 3.0
6 events
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| Oct 19, 2017 at 17:43 | comment | added | Marc Nardmann | @Ali Taghavi: This is off-topic here, but anyway: For $M=\mathbb{CP}^2$, assume there exist $\mathbb{R}$-vector bundles $E_0,E_1$ of rank 2 with $TM=E_0\oplus E_1$. We have $c_1(TM)=3$ and $c_2(TM)=3$, thus $w_2(TM)=1$ and $w_4(TM)=1$; see Milnor/Stasheff 14.10, 14-B. Moreover, $w(E_i)=1+w_2(E_i)$ and $w(TM) = w(E_0)w(E_1)$, thus (in $H^{2k}(M;\mathbb{Z}_2)=\mathbb{Z}_2$) $1=w_2(TM)=w_2(E_0)+w_2(E_1)$ and $1=w_4(TM)=w_2(E_0)w_2(E_1)$, which is impossible. Hence $M$ does not even admit a 2-plane distribution. | |
| Oct 18, 2017 at 6:04 | comment | added | Ali Taghavi | @MarcNardmann What about $\mathbb{C}P^2$?Does it admit a 2 dimensional real foliation? If yes is there such a foliation whose leaves are totally geodesics immersed submanifolds of $\mathbb{C}P^2$ where the later is equiped with the Fubbini - study metric? | |
| Aug 10, 2017 at 12:12 | comment | added | Marc Nardmann | I had overlooked that Q4 has been asked before on mathoverflow. There is a 7-dimensional foliation on $S^{15}$: link. | |
| Aug 9, 2017 at 22:38 | comment | added | Marc Nardmann | @ThiKu: Answering Q1 completely would be a very ambitious research project, but asking for an answer (better than a simple "no") to Q2 or Q3 seems like a legitimate mathoverflow question to me, although it's maybe not an easy one. I am willing to accept an answer with a single example concerning Q2 or Q3. Also, I don't expect that the correct answer to Q4 is "nothing". | |
| Aug 9, 2017 at 19:54 | comment | added | ThiKu | To me this looks more like a research project than like a mathoverflow question :-) | |
| Aug 9, 2017 at 12:04 | history | asked | Marc Nardmann | CC BY-SA 3.0 |