Delta 2 x beta.

Now we want to express this quantity in a bit different way.

Let us introduce, notice that this quantity is antisymmetric,

and the exchange of alpha and beta in this is antisymmetric.

So it changes its sign and the change of alpha and beta [INAUDIBLE].

And let me introduce this guy,

delta S of beta which is the following,

it's delta 1 x alpha delta 2 x beta

minus delta 1 x beta delta 2 x alpha.

This quantity defines the area of that

parallelogram that we have been considering,

A B C, A D C, this quantity.

So, if we introduce this quantity, we can write this expression.

So, we basically can use the antisymmetry of this quantity, we can,

instead of this guy, we can use this guy.

But, we have to, so

this is equal then to 1 minus

one half R mu nu alpha beta

V nu delta S alpha beta.

So, the angle of the rotation of these guys, the result of the rotation of

this vector, after parallel transporting along two different guys.

Is proportional to the area of this parallelogram,

proportional to the vector itself, and

it is proportional to this quantity.

Which has the following form, by definition as follows from this formula.

R r mu nu alpha beta is just

d alpha gamma mu, mu,

beta minus d beta gamma mu,

nu alpha plus gamma mu,

gamma alpha,

gamma gamma nu beta.

Minus gamma mu gamma beta,

gamma gamma nu alpha.

So, this quantity is nothing but the Riemann tensor.

This is exactly Riemann tensor.

One frequently uses also a different expression for

it when we use all lower case indices.

So this is just g mu gamma R gamma nu alpha beta.

So lower case indices.

This guy is nothing but curvature for this connection.

That is another interpretation of this guy.

So in flat spacetime,

well first of all, this is a tensor.

So it transforms as a tensor on the coordinate transformation, so

it means multiplicatively.

In flat spacetime, one can choose everywhere Minkowskian metric.

In Minkowskian metric, this is zero so all is zero.

So this guy is zero.

In Minkowskian metric that is obviously zero.

But because it's zero in one coordinate system

which globally covers whole space time.

It is zero in any other coordinate system.

So in flat space, independently on the coordinate system that we use,

this guy is zero, flat space.