Path: blob/main/25Manifold_Torsion_Curvature_Forms.ipynb
Views: 83
25. Torsion and curvature forms
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
To describe connections one can use bases not necessarily induced by coordinate systems. We can use sets of smooth vector fields defined on some open subset of the manifold such that, at each point , the tangent vectors at form a basis of . Let the set of 1-forms be its dual basis (i.e. ). If there exists a coordinate system such that or, equivalently, , we will say that the basis is holonomic. In the present notebook the set will denote not necessarily holonomic basis.
Connection forms
If is a connection on a manifold , the connection forms, , with respect to the basis , are the 1-forms defined by
or equivalently
Example 25.1
On the two-dimensional manifold we have four connection forms :
If we define
then for , we have and
i.e., and according to (25.1')
which is analogous to (21.5) but holds also for non-holonomic bases.
Torsion forms
If is the torsion, then the torsion 2-forms with respect to are defined by
or equivalently
Since is antisymmetric, each is 2-form.
Example 25.2
On the two-dimensional manifold we have two torsion 2-forms:
First Cartan structure equation
Assume we have vector fields and From (21.14) it follows
for 1-forms on , so
so
Using (*) once again we obtain
From (16.6) we know that
and from the definition of the exterior product
From those facts and (25.7) we obtain
We have proved the first Cartan structural equation
Curvature forms
If is the curvature, then the curvature forms with respect to the basis , are defined by
or equivalently
Example 25.3
On the two-dimensional manifold we have four curvature forms:
Second Cartan structural equations
If we use the formulas
then we can check that
We have proved the second Cartan structural equation
Observation: If is an antisymmetric matrix, then for 1-forms on and
If the components of the torsion and the curvature tensors with respect to the basis are defined by then
In fact and
Example 25.4
Consider the half-plane with the connection coefficients with respect to the bases equal to and all other coefficients equal to zero.
Connection forms can be computed using the formula (25.3), :
Torsion forms can be computed from the first Cartan structure equation (25.8), :
$$\theta^1=de^1+\omega^1_1\wedge e^1+\omega^1_2\wedge e^2 =d(dx)-\frac{1}{y}dy\wedge dx-\frac{1}{y}dx\wedge dy=0,\\ \theta^2=de^2+\omega^2_1\wedge e^1+\omega^2_2\wedge e^2 =d(dy)+\frac{1}{y}dx\wedge dx-\frac{1}{y}dy\wedge dy=0.\\$$The second Cartan structure equation (25.10): can be used in calculation of curvature forms:
Since by (25.11) , and is antisymmetric in , then
Using an analogous calculation for , we see that the different from zero components of the curvature tensor are equal to:
.
Example 25.5
Consider the plane with Christoffel symbols and the remaining symbols equal to 0.
Compute connection forms:
and torsion forms:
Now let us try to compute curvature forms.
Before we show the results, we have to do some simplifications.
Example 25.6
In previous examples, the torsion forms were equal to zero.
Let us consider a 2-dimensional manifold with "random" connection coefficients.
Compute the connection forms:
Recall that the torsion was defined in (21.13) as
In SageMath the torsion
method returns the tensor of type (1,2) defined by
The torsion forms in our example are equal to:
and the curvature forms: