hey Guys,I answered This question (math.stackexchange.com/questions/2090665/…) and then Barto gave a link of an exact question, but it do not have any answer.. Should I answer that question too
Usually one references a duplicate that was asked and answered, not one that never received one.
I don't know what the standard line is, but I wouldn't be surprised if the older one would be the one to close on the grounds that it never got an answer.
But I dunno. The Math Mod's office might have a more definitive standard.
Hmm... I'm personally more in favour of meta, as 1) this isn't exactly an urgent question, and 2) it's more clearly documented for anyone with the same question at a later time.
I think the key is that when a question is closed as a duplicate, the link does not say "This question has already been asked here" but rather "This question already has an answer here"
The differential geometry language comes from you f***ed-up physicists ... looking at transformation of components. For mathematicians, $T$ is covariant and $T^*$ is contravariant :P
So you have to remember the interpretation of the tangent space (similarly, cotangent space) of $\Bbb P^3$ at a point $x$. $x$ is a line, and any tangent vector corresponds to point in the orthogonal complement of that line.
"A point of $\Bbb P(T^*\Bbb P^3)$ corresponds to a line $\ell$ and a plane through $\ell$ which, as a tangent plane in $\ell$, is annihilated by a (line's worth of) covector(s) in $T^*_\ell\Bbb P^3$." Let's call that covector $\alpha$. This makes perfect sense to me... Except that $\ker\alpha$, to my mind at least, naturaly lies in $T_\ell \Bbb P^3\cong \Bbb C^3$... Also, it doesn't contain $\ell$ in a natural way.
@s.harp: I can't remember a time I downvoted without posting a comment to complain. And usually I give the person time to fix what I complain about. When no one does, then I downvote. But I am not a typical citizen.
I was actually asking you guys for input on how to make the answer better. If you have none, so be it.
@TedShifrin Like for a sphere. Okay, I'm fine with this. Now for the tangent bundle. Here we should get a 2-dimensional subspace (not a 3-dim one) containing $\ell$... I'm tempted to just say $\Bbb C\ell\oplus \Bbb C v$ where $v$ is a tangent vector.
@TedShifrin I'm telling you, Hirzebruch says we should understand the tangent bundle in terms of a flag $(1)\subset (2)\subset (4)$. So we're looking for a plane containing $\ell$ in this case.
the proof I most enjoyed was $\sum \frac{1}{n^2}$ is convergent, via comparing the terms with terms from $\frac{1}{2^x}$. The diagonal argument from cantor is another I liked very much
@Ted Maybe I would emphasize more on the fact that convergence is an asymptotical property, thus if two series "tend to be the same" at $\infty$, they have similar nature (is that the terminology). Otherwise your answer seems absolutely fine to me.
@TedShifrin Yeah, I definitely think that works (you'll get a scaling factor, as we discussed for the Euler sequence, but that doesn't mess with the span)
