Connection matrix in orientable 2-bundle is skew-symmetric
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In the notes I am following to learn about connections, there is the following lemma:
whose proof is natural and I understand. Later in the text the author writes the following (referring to a metric connection):
I suppose this comes from the lemma above, but I do not see how. Is $nabla$ somehow a metric connection just because we are dealing with an orientable bundle?
differential-geometry vector-bundles connections
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
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add a comment |
$begingroup$
In the notes I am following to learn about connections, there is the following lemma:
whose proof is natural and I understand. Later in the text the author writes the following (referring to a metric connection):
I suppose this comes from the lemma above, but I do not see how. Is $nabla$ somehow a metric connection just because we are dealing with an orientable bundle?
differential-geometry vector-bundles connections
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
$endgroup$
1
$begingroup$
The written equality does certainly not hold for a general connection. You do not specify in your post the nature of your connection, but by the context, it seems that it is compatible with the metric.
$endgroup$
– Amitai Yuval
Dec 29 '18 at 12:05
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@AmitaiYuval yes, I will add that.
$endgroup$
– Soap
Dec 29 '18 at 12:10
add a comment |
$begingroup$
In the notes I am following to learn about connections, there is the following lemma:
whose proof is natural and I understand. Later in the text the author writes the following (referring to a metric connection):
I suppose this comes from the lemma above, but I do not see how. Is $nabla$ somehow a metric connection just because we are dealing with an orientable bundle?
differential-geometry vector-bundles connections
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
$endgroup$
In the notes I am following to learn about connections, there is the following lemma:
whose proof is natural and I understand. Later in the text the author writes the following (referring to a metric connection):
I suppose this comes from the lemma above, but I do not see how. Is $nabla$ somehow a metric connection just because we are dealing with an orientable bundle?
differential-geometry vector-bundles connections
differential-geometry vector-bundles connections
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
edited Dec 29 '18 at 12:10
Soap
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
asked Dec 29 '18 at 11:56
SoapSoap
1,027615
1,027615
1
$begingroup$
The written equality does certainly not hold for a general connection. You do not specify in your post the nature of your connection, but by the context, it seems that it is compatible with the metric.
$endgroup$
– Amitai Yuval
Dec 29 '18 at 12:05
$begingroup$
@AmitaiYuval yes, I will add that.
$endgroup$
– Soap
Dec 29 '18 at 12:10
add a comment |
1
$begingroup$
The written equality does certainly not hold for a general connection. You do not specify in your post the nature of your connection, but by the context, it seems that it is compatible with the metric.
$endgroup$
– Amitai Yuval
Dec 29 '18 at 12:05
$begingroup$
@AmitaiYuval yes, I will add that.
$endgroup$
– Soap
Dec 29 '18 at 12:10
1
1
$begingroup$
The written equality does certainly not hold for a general connection. You do not specify in your post the nature of your connection, but by the context, it seems that it is compatible with the metric.
$endgroup$
– Amitai Yuval
Dec 29 '18 at 12:05
$begingroup$
The written equality does certainly not hold for a general connection. You do not specify in your post the nature of your connection, but by the context, it seems that it is compatible with the metric.
$endgroup$
– Amitai Yuval
Dec 29 '18 at 12:05
$begingroup$
@AmitaiYuval yes, I will add that.
$endgroup$
– Soap
Dec 29 '18 at 12:10
$begingroup$
@AmitaiYuval yes, I will add that.
$endgroup$
– Soap
Dec 29 '18 at 12:10
add a comment |
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$begingroup$
The written equality does certainly not hold for a general connection. You do not specify in your post the nature of your connection, but by the context, it seems that it is compatible with the metric.
$endgroup$
– Amitai Yuval
Dec 29 '18 at 12:05
$begingroup$
@AmitaiYuval yes, I will add that.
$endgroup$
– Soap
Dec 29 '18 at 12:10