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I quote http://en.wikipedia.org/wiki/Torsion_tensor#Affine_developments":

I try to apply this to the natural connection on the tangent bundle of M = S

I mean here natural connection the connection which defines the parallel transport so that

1. Tangent vectors on the Equator pointing toward North are parallel transported along the Equator into each other.

2. The longitude lines are geodesics, i.e. tangent vectors of a given longitude line are parallel transported along this longitude line into each other.

Now I take x

Applying the definition of Wikpedia, the affine development of this loop is the following open polygon in T

If this is so, then according Wikipedia's definition, this connection has a non-vanishing torsion. But in this case, this connection can't be a Levi-Civita connection because the Levi-Civita connection is torsion-free by definition. Is this really so? The connection defined above is really not a Levi-Civita connection, or I have misunderstood or miscalculated something?

Suppose that x_{t}is a curve in M. The affine development of x_{t}is the unique curve C_{t}in T_{x0}M such that

[tex]\dot{C}_t = \tau_t^0\dot{x}_t,\quad C_0 = 0[/tex]

where

[tex]\tau_t^0 : T_{x_t}M \to T_{x_0}M[/tex]

is the parallel transport associated to ∇.

In particular, if x_{t}is a loop, then C_{t}may or may not also be closed depending on the torsion of the connection.

I try to apply this to the natural connection on the tangent bundle of M = S

^{2}(or more intuitively, of the surface of the Earth)I mean here natural connection the connection which defines the parallel transport so that

1. Tangent vectors on the Equator pointing toward North are parallel transported along the Equator into each other.

2. The longitude lines are geodesics, i.e. tangent vectors of a given longitude line are parallel transported along this longitude line into each other.

Now I take x

_{t}a triangle on the sphere having three rectangles : x_{0}= (0,0), x_{π/2}= (0, π/2), x_{π}= (π/2,π/2) = (π/2,0), x_{3π/2}= x_{0}. (with (lat, long) coordinates)This is a loop.Applying the definition of Wikpedia, the affine development of this loop is the following open polygon in T

_{(0,0)}S^{2}: C_{0}= (0,0), C_{π/2}= (0, π/2), C_{π}= (π/2, π/2), C_{3π/2}= (0, π/2).If this is so, then according Wikipedia's definition, this connection has a non-vanishing torsion. But in this case, this connection can't be a Levi-Civita connection because the Levi-Civita connection is torsion-free by definition. Is this really so? The connection defined above is really not a Levi-Civita connection, or I have misunderstood or miscalculated something?

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