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Sep
10
comment How to show the height function of a torus has 4 critical points
Thanks, @t.b.! I will take a look.
Sep
10
asked How to show the height function of a torus has 4 critical points
Aug
1
accepted generalized Rayleigh Quotient
Aug
1
comment generalized Rayleigh Quotient
Thank you! I didn't check it in detail, but looks right to me. And about that change of scalar product, that's similar to the answer Will Jagy gave. but since you are the only one answering the problem in a direct way, i will pick your answer as the best one. Thank you!
Aug
1
comment generalized Rayleigh Quotient
why is that for any $y\in A$, $y$ can be written as $y=h-(h,f)f$, while your $h$ here in any vector in $X$?
Jul
27
accepted Proof of Gauss's Lemma (Riemannian Geometry version)
Jul
27
comment Proof of Gauss's Lemma (Riemannian Geometry version)
Now that you confirmed what I wrote was right, there's definitely some mistake in Wiki. Thanks a lot. For beginners like me, some times it's hard to tell right from wrong.
Jul
27
comment Proof of Gauss's Lemma (Riemannian Geometry version)
I think the result should be the parallel transport of $v$ along the geodesic at the point $\exp_p(v)$. This makes sense, and it yields what I need. Because Parallel transportation preserves the inner product if the connection is compatible with the metric.
Jul
27
comment Proof of Gauss's Lemma (Riemannian Geometry version)
@ZhenLin Thank you for the clarification!
Jul
27
comment Proof of Gauss's Lemma (Riemannian Geometry version)
I assume you mean $\exp_p (tw_T)$. But I would like to know why the method used in Wiki is also right. By the definition of the differential of a map, to compute $(d\,\exp_p)_v(v)$, one construct a curve $\alpha(t)$, with $\alpha(0)=v$(the point where the differential is evaluated) and $\alpha'(0)=v$ (the direction). So a candidate for such $\alpha(t)$ would be $\alpha(t)=(t+1)v$. Now, by definition, $$(d\,\exp_p)_v(v)={d\over dt}(\exp_po \alpha(t))|_{t=0}={d\over dt}\gamma(1,p,(t+1)v)|_{t=0}={d\over dt}\gamma(t+1,p,v)|_{t=0}$$And then, how does it conclude that the result is $v$?
Jul
27
asked Proof of Gauss's Lemma (Riemannian Geometry version)
Jun
8
comment To define a measure, is it sufficient to define how to integrate continuous function?
This is a really nice answer, Nate! Thank you.
Jun
8
accepted To define a measure, is it sufficient to define how to integrate continuous function?
Jun
8
asked To define a measure, is it sufficient to define how to integrate continuous function?
May
28
accepted Prove the equation has a root.
May
28
comment Prove the equation has a root.
I see! Why didn't I realize that before!! Thank you so much!
May
28
asked Prove the equation has a root.
May
28
accepted Can anybody recommend me a topology textbook?
May
28
comment Can anybody recommend me a topology textbook?
As you are the only person who answered, I will pick your answer. Thank you for all of you who commented as well.
May
28
comment Can anybody recommend me a topology textbook?
@William Is the book you recommend about point set topology or algebraic topology? or it covers both?