What Are the Key Concepts in Alexei Kovalev's Article on Vector Subspaces?

[math6]

hello friends,
i have some point that i didn't understand when i read the article of ALEXEI KOVALEV, if you can help me to answer to this question: we can found the article in this lien (
www.dpmms.cam.ac.uk/~agk22/vb.pdf )

1- why a vector from vector subspace must be spanned just by {∂/(∂a^i )}
2-why does he mean when he said that ( θ_p^i )cannot all vanish on a vertical vector.
3-why we must have [ f_k^i (p) ] and [ g_k^i (p)] smooth to define a field of horizantal subspaces
4- i didn't understand the linearity of function [e_k^i (p) ] " are linear in the fibre variables.

thnx very much.

 

[arkajad]

Your questions are so basic that they indicate holes in your understanding. I would suggest you first study some basic text on differential geometry, manifolds, tangent vectors, forms, best with exercises. Did you study one? Which one?

 

[math6]

yes I studied differential geometry for 3 months but only during free periods, we have not time to work properly. I would like really do exercises on these themes because only with exercises that we can master our information.
And now when I am preparing my brief I found great difficulty, if you can help me when i can found exercises to help me to train .
i will be really so greatful.

 

[arkajad]

I would suggest the book by Marion Fecko http://books.google.fr/books?id=vQR..._result&ct=result&resnum=8&ved=0CD4Q6AEwBzge". But really you should visit a library and get a book that suits you best. Once you are thoroughly done with the foundations - the rest will come easy. But you need one good book. Some, like "Tensor analysis on Manifolds" by Bishop and Goldberg you can get for yourself quite cheap.

 

[math6]

thnx arkajad ; but have you any idea for the answers to the question ?
i have an little exams and i have to finich this article before Tuesday .
thnx again .

 

[arkajad]

OK.
"1- why a vector from vector subspace must be spanned just by {∂/(∂a^i )}"

Probably you mean "vertical subspace". a^i are coordinates in m-dimensional fibres provided by a local trivialization. Then \frac{\partial}{\partial a^i} are vector fields (or vectors, if you take them at a point) tangent to the coordinate lines. For any manifold vectors tangent to coordinate lines span the tangent subspace. This follows from the definition of the tangent subspace and tangent vectors. Here, in particular, we apply this general property to the fibre.

"2-why does he mean when he said that ( θ_p^i )cannot all vanish on a vertical vector."

This is an assumption, not a statement. He wants to take ( θ_p^i ) with such a property. In other words: he wants the kernel of ( θ_p^i ) to be transversal i.e. no common element except 0, to the fibre.

"4- i didn't understand the linearity of function [e_k^i (p) ] " are linear in the fibre variables."

That means e_k^j(x,a) must have the property

e_k^j(x,\lambda a+\mu a')=\lambda e_k^j(x,a)+\mu e_k^j(x,a'),\, (\lambda,\mu\in R), what implies that they must be of the form:
e_k^i(x,a)=\Gamma_{jk}^i(x)a^j.

 

[math6]

For the first question I do not agree.Excuse me but all the vector space tangent T_p E must be written in the basis {∂ / (∂ x ^ k), ∂ / (∂ a ^ i)}, so according to the definition of vertical subspace, this is the set of vectors X of T_p E which verifies dπ (X) = 0.

so ALEXEI concludes that these are just the vector which is written in the base{∂ /(∂ a ^ i)}.
So why it must be only in the database to verify that dπ (X) = 0.
That was my question.
For the question 2 and 3, I am almost agree ..

 

[arkajad]

For the first question I do not agree.Excuse me but all the vector space tangent T_p E must be written in the basis {∂ / (∂ x ^ k), ∂ / (∂ a ^ i)}, so according to the definition of vertical subspace, this is the set of vectors X of T_p E which verifies dπ (X) = 0.

Coordinates x^k are constant along the fibers. Therefore only ∂ / (∂ a ^ i) are needed to span the vertical tangent subspace.

 

[math6]

could you give me some reference book where I can find more examples on the vector subspace and the horizontal subspace?

 

[arkajad]

I suggest you start a new thread, explaining how much you know and asking for an advice there. I could not find anything that will suit your level in a short time.