Recent content by Mehdi_

Christoffel Symbols from the metric tensor By definition the metric tensor g_{ij} is : g_{ij} = \frac{\partial z^n}{\partial x^i} \frac{\partial z^n}{\partial x^j} Therefore the derivative of g_{ij} is : g_{ij} = \partial_i z^n \partial_j z^n \partial_k g_{ij} =...

Christoffel Symbols An affine connection in the case of a Riemannian manifold is a Levi-Civita connection if it preserves the metric and is Torsion-free. The components of this connection with respect to a system of local coordinates are then called Christoffel symbols. There are two...

Covariant derivatives & differential Let e_i be local basis and x^i be generalized coordinates. The differential of a vector \vec{v} = v^i e_i is : d \vec{v} = d(v^i e_i) = e_i dv^i + v^i de_i But d \vec{v} = \frac{\partial \vec{v}}{\partial x^j} dx^j therefore...

Covariant derivatives & vectors The covariant derivative of a scalar field or a function is : \nabla_i \varphi = \frac{\partial \varphi}{\partial x^i} And the covariant derivative of a vector \vec{v} = v^i e_i is just : \nabla_i \vec{v} = \frac{\partial \vec{v}}{\partial x^j}...

Metric & line element The coordinates (contravariant components) of a point in a coordinate system are written x^i. The radius vector of the point is then : r = x^k e_k Let ds be the arc length between two close points x^i and x^i + dx^i . And let the vector dr joining the...

Tensors & metric contravariant components : A = A^1 e_1 + A^2 e_2 + A^3 e_3 = A^i e_i covariant components : A = A_1 e^1 + A_2 e^2 + A_3 e^3 = A_i e^i Therefore : A = A_i e^i = A^i e_i and then (A^i e_i) \bullet e^k = (A_i e^i) \bullet e^k A^i (e_i \bullet e^k) = A_i...

Inner product and metric tensor Recall that the scalar product of two vector is v.w : v.w = (v^{\alpha} e_{\alpha}) (w^{\beta} e_ {\beta}) = v^{\alpha} w^{\beta} (e_{\alpha} . e_{\beta}) = v^{\alpha} w^{\beta} \eta _{\alpha \beta} (e_{\alpha} . e_{\beta}) = \eta_{\alpha \beta}...

Complex numbers & orthogonal matrices A complex number is written in the form z=a+ib where a and b are real numbers while i is a symbole which satisfy i^2=-1. In polar coordinates, z=r(cos(\theta)+isin(\theta)) where r is the magnitude and \theta is the angle. However complex numbers...

The metric g_{ij} = J^T \ J of the sphere : x = cos(\theta)sin(\phi) y = sin(\theta)sin(\phi) z = cos(\phi) The Jacobian J is : J = \left[ \begin {array}{ccc} \cos \left( \theta \right) \sin \left( \phi \right) &-r\sin \left( \theta \right) \sin \left( \phi \right) &r \cos...

Holonomic bases A holonomic basis for a manifold is a set of basis vectors e_k for which all Lie derivatives vanish: [e_j, e_k] = 0 Given coordinates x^a, we define basis vectors e_a and basis one forms \omega^a in the following way: e_a = \partial_a= \frac{\partial }{\partial x^a}...

The metric g_{ij} of the torus above could also be computed by the formula : g_{ij} = J^T \ J where J denotes the Jacobian and J^T its transpose. If the torus can be defined parametrically by : x = (c + a \ cos(v) ) \ cos(u) y = (c + a \ cos(v) ) \ sin(u) z = a \...

The Line Element and Metric of a torus If the major radius of this torus is c and the minor radius a ; with c>a . The torus S(u,v) can be defined parametrically by: x = (c + a \ cos(v) ) \ cos(u) y = (c + a \ cos(v) ) \ sin(u) z = a \ sin(v) where u and v \in [0, 2 \pi ]...

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 ...

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

What is then the real use Cartan's first structure equation and the spin connection in what is above ?