Koszul Connexion: Explained Simply

  • Thread starter Calabi
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In summary, A Koszul connexion is a mathematical concept used in vector bundle theory. It is used to associate a vector field and a section with another section on a vector bundle. The French definition is preferred by some, but the English version is clearer and more specific. The function, f, is defined on the base or manifold, and is generally not a global section of the vector bundle. However, it can be made zero outside of a neighborhood, making it non-zero and therefore not a concern.
  • #1
Calabi
140
2
Hell every body, do you know what is a Koszul connexion please? WIkipédia francais is not clear
about that for me.

I just know if I have a vectorial bundle [tex]\pi : E \rightarrow B[/tex], a vectorial field X on B and a section S on B. A connexion associate at those 2 object an other section.

But it's nt clear.

Thank you in advance and have a nice afternoon:oldbiggrin:.
 
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  • #2
Calabi said:
Hell every body, do you know what is a Koszul connexion please? WIkipédia francais is not clear
about that for me.

I just know if I have a vectorial bundle [tex]\pi : E \rightarrow B[/tex], a vectorial field X on B and a section S on B. A connexion associate at those 2 object an other section.

But it's nt clear.

Thank you in advance and have a nice afternoon:oldbiggrin:.

Did you read the English Wikipedia article? It is very clear.

http://en.wikipedia.org/wiki/Connection_(vector_bundle)
 
  • #3
I prefere the french formulation(whitout the tensor for the moment.). than the english. But the french is not clear. On the french version what is f please?
 
  • #4
Calabi said:
I prefere the french formulation(whitout the tensor for the moment.). than the english. But the french is not clear. On the french version what is f please?
send me the link
 
  • #7
Calabi said:
I prefere the french formulation(whitout the tensor for the moment.). than the english. But the french is not clear. On the french version what is f please?

Calabi, f is a smooth function. X is a tangent vector at a point on the manifold. Note that s is a section of a vector bundle - but not necessarily the tangent bundle.

Can you ask some specific questions. I do no want to just repeat what the Wikipedia article is saying.
 
  • #8
The thing is they don't specifise the definition set and the arrived set of f it's just what's stike me.
That's why I'm wooriing of. Excuse me if you you think I want you to repaet what wikipédia said.
 
  • #9
Calabi said:
The thing is they don't specifise the definition set and the arrived set of f it's just what's stike me.
That's why I'm wooriing of. Excuse me if you you think I want you to repaet what wikipédia said.

Milles pardons Pour moi, l'article de Wikipedia est précis et sans des questions spécifiques je ne sais pas comment l'expilquer d'une autre manière. (Pardonnez mon mauvais français).

La fonction,f , est défini sur la base,la variété, meme comme le champs de vecteurs.

En general il n'existe pas une section globale d'un fibre vectoriel sans des zeros mais c'est toujours possible de l'éloigner a travers tout le variété en le faisant zero a l'extérieur d'un voisinage ouvert. En ce cas, on ne s'inquiet pas aux zeros mais seulement au voisinage non-zero.
 

What is a Koszul Connexion?

A Koszul Connexion is a mathematical concept that describes a way to connect two vector spaces. It is used in the field of differential geometry to study the geometric properties of manifolds.

How is a Koszul Connexion defined?

A Koszul Connexion is defined as a bilinear map that takes in two vector fields and outputs another vector field. It satisfies certain properties, such as being linear in each of its inputs and satisfying the Leibniz rule.

What is the significance of a Koszul Connexion?

A Koszul Connexion allows us to define a notion of differentiation on a manifold that does not have a natural metric or coordinate system. This allows us to study the geometry of manifolds in a coordinate-free way.

How is a Koszul Connexion related to Riemannian geometry?

A Koszul Connexion is a generalization of the Levi-Civita connexion, which is the unique connexion that preserves the metric structure on a Riemannian manifold. This means that a Koszul Connexion reduces to the Levi-Civita connexion when the manifold has a Riemannian metric.

What are some applications of Koszul Connexions?

Koszul Connexions have applications in various fields such as physics, computer science, and machine learning. They are used in theories of gravity, such as general relativity, and in the study of differential equations and optimization problems. They also play a role in developing algorithms for data analysis and machine learning tasks.

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