Geometry of symmetric spaces and Lie groups in physics

[Mehdi_]

How can we write the spin connection (and the Cartan's first structure equation ) if the manifold is a simple torus ?

same question if the manifold now is $$z=5-x^2-y^2$$

 

[garrett]

How can we write the spin connection (and the Cartan's first structure equation ) if the manifold is a simple torus ?

0

same question if the manifold now is $$z=5-x^2-y^2$$

Ah, an embedded surface. That's going to take a couple of pages of algebra to work out the frame and connection. I encourage you to tackle it on a different thread. ;) I'll give you a hint though: treat this manifold as a parameterized surface, embedded in 3D, with coordinates (parameters) x and y. It's going to take some work to get the coefficients of two orthonormal tangent vectors over this surface,
$$\vec{e_1} = a \vec{\frac{\partial}{\partial x}} + b \vec{\frac{\partial}{\partial y}}$$
$$\vec{e_2} = c \vec{\frac{\partial}{\partial x}} + d \vec{\frac{\partial}{\partial y}}$$
, then more work to get the frame, and more to get the connection. Maybe start by choosing c=0, and solve for a,b,c.

I'll look in on the other thread and see how you're doing.

 

[Mehdi_]

Question 1: To calculate the spin connection of $$z=5-x^2-y^2$$ could we calculate first :
The metric, Ricci Rotation coefficients, christoffel symbols, those orthonomal basis (why not nonholonomic basis?), Riemann tensor, Ricci tensor, Ricci scalar, tetrad method and curvature one forms ??

Question 2: Why it is so important to know the curvature ? Does the spin connection (or maybe Cartan's first structure equation) give information about the curvature ?
Probably Riemann curvature tensor does ?...
What is the relation between the spin connection and Riemann curvature tensor ?

Question 3: Does the curvature give the strength of the field ?