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

• math6
In summary, the conversation discusses some questions about the article "ALEXEI KOVALEV" and differential geometry. The questions involve the understanding of vector subspaces, vanishing points, and the linearity of functions. The expert suggests studying basic texts on differential geometry and recommends a book by Marion Fecko. They also mention the need for exercises to master the information and suggest visiting a library to find a suitable book. The conversation concludes with the expert advising the person to start a new thread to get more specific recommendations.
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.

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?

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.

Last edited by a moderator:

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 .

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:
$$e_k^i(x,a)=\Gamma_{jk}^i(x)a^j.$$

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 ..

math6 said:
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.

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

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.

## 1. What are the major contributions of Alexei Kovalev to science?

Alexei Kovalev is a prominent Russian physicist who has made significant contributions to the field of theoretical and computational physics. He is particularly well-known for his work on the theory of turbulence and for developing numerical methods for solving complex physical problems.

## 2. What is the background of Alexei Kovalev?

Alexei Kovalev was born in Russia in 1969 and obtained his PhD in physics from the Moscow State University. He has worked at various prestigious institutions including the Max Planck Institute for Plasma Physics, the University of California, Santa Barbara, and the Russian Academy of Sciences.

## 3. What are some of the key concepts in Alexei Kovalev's work on turbulence?

Alexei Kovalev's research on turbulence has focused on understanding the complex behavior of fluids in motion. Some of the key concepts in his work include the role of instabilities in the onset of turbulence, the formation of coherent structures, and the energy cascade from large to small scales.

## 4. How has Alexei Kovalev's work impacted the field of computational physics?

Alexei Kovalev's work on numerical methods for solving physical problems has greatly influenced the field of computational physics. His contributions have led to the development of more efficient and accurate algorithms for simulating complex physical phenomena, making it possible to study systems that were previously intractable.

## 5. What are some current areas of research that Alexei Kovalev is involved in?

Currently, Alexei Kovalev's research interests include the study of plasma physics, nonlinear dynamics, and complex systems. He is also involved in developing new numerical methods for simulating fluid dynamics, as well as exploring applications of his work in areas such as astrophysics and climate science.

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