Questions on connections and covariant differentiation

[kakarotyjn]

The question is in the pdf file,thank you!




M is a Riemannian manifold, $\vdash$ is a global connection on M compatible with the Riemannian metric.In terms of local coordinates $u^1,...,u^n$ defined on a coordinate neighborhood $U \subset M$, the connection $\vdash$ is determined by $\Gamma^k_{ij}$ on U,as follows. Let $\partial_k$ denote the vector field $\frac{\partial}{\partial u^k}$ on U.Then any vector field X on U can be expressed uniquely as $$X=\sum_{k=1} x^k \partial_k$$ where the $x^k$ are real valued functions on U.In particular the vector field $\partial_i \vdash \partial_j$ can be expressed as $$\partial_i \vdash \partial_j=\sum_k \Gamma^k_{ij}\partial_k$$

My 1st question is that there is an equation $$\partial_i g_{jk}=<\partial_i \vdash \partial_j,\partial_k>+<\partial_j,\partial_i\vdash \partial_k>$$,but how could it be? The left of the equation is a vector field but the right is a function.And there are the first Christoffel identity $$<\partial_i\vdash\partial_j,\partial_k>=\frac{1}{2}(\partial_ig_{jk}+\partial_jg_{ik}-\partial_kg_{ij})$$
and the second Christoffel identity $$\Gamma^l_{ij}=\sum_k\frac{1}{2}(\partial_ig_{jk}+\partial_jg_{ik}-\partial_kg_{ij})$$ for which I have the same question.How could a vector field equals a real function?

My 2nd question is why in Euclidean n-space,$R^n$,have $$\Gamma^k_{ij}=0?$$ I use the second Christoffel identity but obviously it don't equals zero.The metric is the usual Riemannian metric $dx_1^2+dx_2^2+...+dx_n^2$

 

[quasar987]

g_jk is a function of U, so how could its derivative wrt to x_i be a vector field? It is a function still.

Equally misterious to me is why you think the Christ. symbols are vectors. Like you wrote before, they are merely the coefficients(=smooth functions on U) of the vector field \partial_i \vdash \partial_j wrt to the basis (\partial_k)_k.

Your second question kinda proves that there's some fundamental misconception you're having about the meaning of some of the symbols. Because indeed you get the vanishing of the Christ. symbols from the 2nd identity. In the euclidean riemannian metric, the metric coefficients g_ij (=smooth R-valued functions on R^n!) are just the constant function g_ii=1 and g_ij=0 (if i is not j). So their derivatives are indeed all 0.

 

[kakarotyjn]

Oh,I get it! $$\partial_i g_{jk}$$ is only the derivative of g_{jk} in the direction of x_i,it is still a function. It was really a misconception I have.

Thank you quasar987!:)

 

[quasar987]

My pleasure. :)

 

[blue_raver22]

I would like to address the questions and concerns raised in this content. First, let us define some terms for clarity. A Riemannian manifold is a mathematical space that is equipped with a Riemannian metric, which is a way to measure distances and angles on the manifold. A global connection on a Riemannian manifold is a way to connect two tangent spaces at different points on the manifold. This connection is compatible with the Riemannian metric, meaning that it preserves the metric's properties.

To answer the first question, the equation $\partial_i g_{jk}=<\partial_i \vdash \partial_j,\partial_k>+<\partial_j,\partial_i\vdash \partial_k>$ is a result of the compatibility of the connection with the Riemannian metric. This equation is known as the Koszul formula and it relates the partial derivatives of the metric with the connection coefficients. The left side of the equation is a vector field, while the right side is a function. This is because the connection coefficients $\Gamma^k_{ij}$ are functions on the manifold, while the vector fields $\partial_i$ and $\partial_j$ are operators that act on functions.

For the second question, in Euclidean n-space, the connection coefficients $\Gamma^k_{ij}$ are indeed zero. This is because the Euclidean space has a flat metric, meaning that the metric is constant and does not vary with position. In this case, the second Christoffel identity simplifies to $\Gamma^l_{ij}=0$. However, in a general Riemannian manifold, the metric can vary with position, leading to non-zero connection coefficients.

In summary, the equations and identities mentioned in the content are fundamental concepts in Riemannian geometry and are derived from the compatibility of the connection with the Riemannian metric. The vector fields and functions involved in these equations have different roles and should not be confused. The second question highlights the importance of understanding the properties of the manifold and its metric in order to fully understand the behavior of the connection and its coefficients.