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Jianbing_Shao's latest activity

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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Yes, I think we should treat the metric compatible equation as differential equations. To do so we at least can understand two problems...

Sep 6, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Simply from a tetrad field , then we can calculate the difference between teleparallel connection and Levi-Civita connection: If we...

Sep 5, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Another problem ,Do you think a non trivial metric field can also be compatible with a zero curvature connection. they call it...

Sep 5, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Then return to the question we have discussed. From a non-zero curvature Levi-Civita connection and metric compatible equation, then how...

Sep 5, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

In your definition, the fields ##e,\Gamma## is chosen independently. But if we demand that the connection ##\Gamma## to be the...

Sep 4, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Yes, I really think there is contradiction here, In the paper I quoted, He claimed that the holonomy group generated from a metric...

Sep 4, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Metric compatible equations and the equations you gave are two differential equations , In fact your equations is more complex, The...

Sep 4, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

Because Levi-Civita connection is also compatible with metric. So it also satisfy the equation: ##\partial_a {e_b}^{I} = {\Gamma^c}_{ab}...

Sep 3, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

I have read an interesting paper 'Conditions on a Connection to be a Metric Connection' by B. G. Schmidt(Commun. math. Phys. 29, 55—59...

Sep 3, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

When we say the teleparallel connection ##\Gamma^{\alpha}_{\mu\nu} =e_a^{\alpha}\partial_{\mu}e^a_{\nu}## is flat , it has vanishing...

Sep 1, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

To the equation above, I have a question: Because the equation above is a differential equations,, So even for equations like...

Aug 31, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

I read some papers about teleparallel connection. I have a question: When we introduce spin connection ##\omega^a_{\hphantom{a}\mu...

Aug 29, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

There are some mistakes, I made a revision. If we start from a zero curvature teleparallel connection ##{\Gamma^c}_{ab} = {e^c}_{I}...

Aug 14, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

In fact, if we start from a zero curvature teleparallel connection ##{\Gamma^c}_{ab} = {e^c}_{I} \partial_a {e_b}^{I}## and the...

Aug 13, 2025
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Jianbing_Shao replied to the thread I A sufficient condition for integrability of equation ##\nabla g=0##.

I have a question : If all Levi-Civita connections compatible with a non-flat metric field only can have non-zero curvature. and we...

Aug 13, 2025

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