Recent content by Sina

erm yes smooth it should be to relate it to some triangulation of the manifold. I was a bit unclear sorry, I think it should be a triangulation of the smooth manifold whose deRham cohomology you want to dualize. I never proved this so I am a bit unsure about the precise conditions.

We have the homology of simplixes (was it called simplical homology?) whose cohomology is precisely isomorphic to deRham cohomology I believe (you can show that each linear functional on the free group of simplices is precisely integration by some form of appropriate degree)

I meant " then dimension of the mth cohomology group is also C(n,m)" not m

edit: finally manged to tex

Hello, I want to ask for a verification of something I did. Let's say I want to compute the m-th cohomology group of the n-torus with coordinates θi. Suppose that I have shown each [dθi_1 \wedgedθi_2 ... \wedge dθi_m] is an independent element of this deRham cohomology (it can be shown by...

Hmm have never seen this way. I usually encountered connection forms for connections on general vector bundles never tried to use them for tangent bundle but it will be good food for taught thanks

Hmm I guess the formula would be simpler because because metric is identity and symbols vanish. But I guess that would not then cover the most general case but only when a=b=1. Your first idea might give another easy answer maybe but my whole aim was to save my self from calculating the...

Hello Sina, You can indeed doing it without resorting to theorema egregium. Let r(u,v) be the surface with line element ds2 = a2du2 + b2dv2 and tangent vectors ru, rv. Then we have the equalities ru.ru = a2 rv.rv = b2 ru.rv =0 Then we have the following ruu.ru = aua ruv.ru =...

You mean r_1 \times r_2 = ε_{ij}^{k} r_{1}^ir_{2}^je_k where e is the basis for R3. Well without knowing what e^k is, I can not calculate the dot product of this with other vectors? The reason why I gave the motivating example is in that case normal vector N is directly related to r and I...

Hello everyone, Let r(u_i) be a surface with i=1,2. Suppose that its first fundamental form is given as ds^2 = a^2(du_1)^2 + b^2(du_2)^2 which means that if r_1 = ∂r/∂u_1 and r_2= ∂r/∂u_2 are the tangent vectors they satisfy r_1.r_2 = 0 r_1.r_1 = a^2 r_2.r_2 =...

Ha I actually never thought of empty sets as such thanks for the enlightening comment :) I will think and research more about it

Yes indeed I meant degree (and Brouwer degree). and yes I later realized disk is with boundary. and if you try to take circle or torus it does not work so everything seems okay =) thanks for the reply

Hello, I have a question regular values and smooth homotopies. Usually in giving the definition of regular value, they disregard the regular values whose inverse image is empty set (although they should be called regular values if we want to be able to say that set of regular values is dense for...

yea I meant gilmore's book when I said many properties are given without proof

A word of caution, Gilmore is not for the mathematically twitchy :p by meaning which you should be able to withstand some amount of informal arguements or should enjoy trying to make them rigorous. The contents of the book looks like a miracle which should also tell you that not everything is...