Riemann Tensor Equation: Simplifying the Riemann-Christoffel Tensor

[redtree]

The Riemann-Christoffel Tensor ($R^{k}_{\cdot n i j}$) is defined as:

$$
R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}
$$

My question is that it seems that the equation can be simplified as follows, and I'm wondering if my understanding is correct or not.

Given following equation for the Christoffel symbol ($\Gamma^{k}_{i j}$):

$$
\Gamma^{k}_{i j} = \textbf{Z}^{k} \frac{\delta \textbf{Z}_{i}}{\delta Z^{j}}
$$Based on this equation, we consider the following term in the Riemann curvature tensor equation

$$

\begin{align}

\Gamma^{k}_{il}\Gamma^{l}_{jn} &= \textbf{Z}^{k} \frac{\delta \textbf{Z}_i}{\delta Z^{l}} \textbf{Z}^{l} \frac{\delta \textbf{Z}_j}{\delta Z^{n}}

\\

&=\textbf{Z}^{k l} \frac{\delta \textbf{Z}_i}{\delta Z^{l}} \frac{\delta \textbf{Z}_j}{\delta Z^{n}}

\end{align}

$$Similarly:

$$

\begin{align}

\Gamma^{k}_{j l}\Gamma^{l}_{i n} &= \textbf{Z}^{k} \frac{\delta \textbf{Z}_j}{\delta Z^{l}} \textbf{Z}^{l} \frac{\delta \textbf{Z}_i}{\delta Z^{n}}

\\

&=\textbf{Z}^{k l} \frac{\delta \textbf{Z}_j}{\delta Z^{l}} \frac{\delta \textbf{Z}_i}{\delta Z^{n}}

\\

&=\textbf{Z}^{k l} \frac{\delta \textbf{Z}_i}{\delta Z^{l}} \frac{\delta \textbf{Z}_j}{\delta Z^{n}}

\end{align}

$$Thus:

$$
\Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}=0
$$If this is true, the Riemann curvature tensor can be simply written as follows:

$$
R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}
$$

Where is my mistake? I'm not sure.

 

[haushofer]

I'm not familiar with your notation, but it seems you use a basis in which the connection vanishes while its derivative does not. This is always possible, but it does not result into a tensor equation since the connections are not tensors.

 

[Orodruin]

What you are doing from (4) to (5) does not seem correct. It is difficult to tell since, as pointed out in #2, you are not following the typical notation.

 

[redtree]

Notation is from Pavel Grinfeld: Introduction to Tensor Analysis and the Calculus of Moving Surfaces
I'm happy to put in different notation; if you could refer a page to me in the notation you prefer, I'm happy to change.
Steps (4) & (5) are really the key point. Why can't the terms be switched?

$\frac{\delta x}{\delta y} \frac{\delta z}{\delta t} = \frac{\delta z}{\delta y} \frac{\delta x}{\delta t}$?

Or similarly:

$\frac{\delta x}{\delta z} \frac{\delta z}{\delta y} = \frac{\delta x}{\delta y} \frac{\delta z}{\delta z}$?

 

[Orodruin]

Why can't the terms be switched?

Why would they be interchangable? They represent different things.

 

[redtree]

I see the mistake; thanks!