Can You Explain the Meaning of the Affine Connection in Simple Terms?

[the shadow]

Hi
I'm looking for a simple definition fo the affine connection because I can't understand it's meanning , that meanning that givse the mathematical formula of the affine connection a life , You could call it the physiacl meanning .
For your knowledge my friend , I'm very good with the calculations of the affine connection and I can calculate it for any metric using the formula :
$$\[ \Gamma _{\alpha \beta }^\mu = \frac{1}{2}g^{\mu \lambda } \left( {\partial _\beta g_{\alpha \lambda } + \partial _\alpha g_{\beta \lambda } - \partial _\lambda g_{\alpha \beta } } \right) \]$$
But the big problem is in it's meanning , so please help me with understanding it .
thanks

 

[the shadow]

where is everybody ?
it doesn't make any sense that no body know the meanning of the affine connection .
is there a problem in the question ?
if there is a problem , let me know anf I'll fix it
if there is no problem , please , answer mt question .
thanks

 

[gel]

By "affine connection" I think you mean http://en.wikipedia.org/wiki/Levi-Civita_connection" .
The $\Gamma^\mu_{\alpha\beta}$ that you have written are the connection coefficients. The connection itself is $\nabla$, a "covariant derivative" operator,
$$\nabla_YX=Y^i(\partial_iX^k+\Gamma^k_{ij}X^j)e_k$$
for a tangent vector Y and vector field X.

This is the unique torsion free connection on the tangent bundle satisfying $\nabla g=0$. Are you familiar with what this means?

 

[the shadow]

Thank you gel for replying
My friend , the word " connection " means that there are two things connected ( or more than two ) , in this case what are the connected things ?
is the connected things are the tangent vector Y and the vectro field X ?
if the answer is ( yes ) , then can you explain more ?
and if the answer is ( no ) ,then what is the answer ?
thanks again .

 

[gel]

I assume the word "connection" is due to the following:
Any connection gives a meaning to parallel transport along a curve. That is, given a smooth curve joining two points P and Q, you can transport any vector at P along the curve to Q in such a way that its "derivative" (calculated using the connection) along the tangent to the curve is 0. So, the connection "connects" the tangent spaces at P and Q.
The condition $\nabla g=0$ just means that parallel transport along a curve preserves the inner product g(X,Y) between any vectors X and Y.

 

[the shadow]

Thanks gel , I understand now ( as I think )
I'll tell you what I understood and you tell me if it's wrong or right :
The connection is a way to connect the tangent space at P to the tangent space at Q (as I think it makes them as one thing ) to keep a vector transports from P to Q pointing in the same direction ( or allow the meanning of the parallel transport )
That's what I understood
Thanks again .

 

[gel]

It doesn't make the tangent spaces at P and Q the same thing. In order to connect the tangent spaces you need two things
1) A connection
2) a curve joining P and Q
so that you can use parallel transport along the curve.
The map between the tangent spaces at P and Q depends on what curve you use. Suppose P and Q are opposite points of a sphere, and you transport a tangent vector along a great circle from P to Q (eg move from the north to south poles along a line of constant longitude). What happen to the vectors depends on which direction you move in.

 

[the shadow]

Thanks gel .
Now I understand . The connection is a way to allow the parallel transport along a curve joining two points P and Q and the parallel transport depends on the curve that we use , so we can understand the connection as an object contains encoded information about the curvature .
I have one more question . I read that the connection coefficients $$\[ \Gamma _{\beta \gamma }^\alpha \]$$ can be thought of as the $$\alpha$$ component of the change in $$\beta$$ due to a parallel transport along $$\gamma$$ .
( I read this http://physics.syr.edu/courses/PHY312.03Spring/keish-walter/connect.htm" )
What does this mean ?
Thanks

 

[gel]

By definition $\Gamma^i_{jk}e_i=\nabla_{e_k}e_j$, so $\Gamma^i_{jk}$ is the e_i - component of the change in e_j as you move in the e_k direction, calculated by the connection $\nabla$. Once you know how the basis vectors e_i vary along the different directions (ie, you know the connection coeffs), then the connection can be reconstructed.

$$\nabla_YX=Y^i\nabla_{e_i}(X^je_j)=Y^i((\partial_iX^j)e_j+X^j\nabla_{e_i}e_j)=Y^i(\partial_iX^k+X^j\Gamma^k_{ji})e_k$$

The only properties of the connection this uses is that $\nabla_YX$ is linear in X and Y, and the product rule $\nabla_Y(\lambda X)=(\partial_Y\lambda)X+\lambda(\nabla_YX)$ for a scalar field $\lambda$.

 

[the shadow]

Thank you gel
I understand now .
Now I have a good understanding of differential geometry ( the bad thing was the connection ) thanks to you .
Thanks again my friend . Thank you very much .

 

[wofsy]

In Euclidean space one can differentiate a vector field by differentiating each of its coordinates. On a manifold a vector field passes through different vector spaces, the various tangent spaces of the manifold, so there is no natural idea of differentiating it. What does it mean to to take a Newton quotient when the vectors are in different tangent spaces?

A connection defines a way to do this. It gives a way to take the derivative of a vector field on a manifold.

By the way, there is no unique connection and so, no single way to differentiate a vector field.

The idea of a connection is easily visualized for a surface in Euclidean 3 space. A vector field on the surface also lives in the ambient Euclidean space so you can differentiate it coordinate wise. Part of its derivative will be tangent to the surface, the other part normal. The connection on the surface is defined to be just the tangential part, the part that a person living on the surface can see.

Notice that one needs an idea of orthogonal projection to get the tangential part. This means that the connection is defined using a metric and so the derivative of the vector field depends on the geometry of the surface.

I recommend a basic book on classical Differential Geometry. There is much intuition in this simpler subject.

It is important to know that the idea of differentiating a vector field does not require the idea of a metric. A connection which is compatible with a metric is special and is called a Riemannian connection . But connections exist without metrics and still define differentiation of vector fields and also produce geometry, just not Riemannain geometry.

Also connections do not have to be on the tangent bundle but can be defined on any vector bundle. It makes sense to differentiate vector fields which are not tangential.

I would be glad to go into this more if you like.

 

[the shadow]

Hi , wofsy
Thank you very much my friend .
I'd love to hear more from you .
And I'd invite you to my new question ( The mystery of geodesics . )
Thanks .

 

[LorenzoMath]

books

if you are studying riemannian geometry, i recommend Manfredo P. do Carmo's Riemannian Geometry. It's a thin book. His definition of connection is axiomatic, but reducing general definitions to down-to-earth definitions or ideas is usually not that hard. It's a nice book.