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https://diffgeom.subwiki.org/w/api.php?action=feedcontributions&feedformat=atom&user=Asafs Diffgeom - User contributions [en] 2026-08-31T23:04:11Z User contributions MediaWiki 1.41.2 https://diffgeom.subwiki.org/w/index.php?title=Tensor_product_of_metric_connections_is_metric&diff=2113 Tensor product of metric connections is metric 2015-09-22T15:18:42Z

<p>Asafs: /* Proof */</p> <hr /> <div>==Statement==<br /> <br /> A [[differential manifold]] <math>M</math>. Two [[fact about::metric bundle]]s <math>(E,g)</math> and <math>(E',g')</math> (i.e., <math>E,E'</math> are [[vector bundle]]s and <math>g,g'</math> are [[Riemannian metric]]s or [[pseudo-Riemannian metric]]s on these). <math>\nabla, \nabla'</math> are [[fact about::metric connection]]s on <math>(E,g)</math> and <math>(E',g')</math> respectively. Then, the [[fact about::tensor product of connections]] <math>\nabla \otimes \nabla'</math> is a metric connection on <math>(E \otimes E',g \otimes g')</math>.<br /> <br /> ==Proof==<br /> <br /> ==Related facts==<br /> <br /> * [[Tensor product of flat connections is flat]]</div>

Asafs https://diffgeom.subwiki.org/w/index.php?title=Tensor_product_of_metric_connections_is_metric&diff=2112 Tensor product of metric connections is metric 2015-09-22T10:04:15Z

<p>Asafs: /* Proof */</p> <hr /> <div>==Statement==<br /> <br /> A [[differential manifold]] <math>M</math>. Two [[fact about::metric bundle]]s <math>(E,g)</math> and <math>(E',g')</math> (i.e., <math>E,E'</math> are [[vector bundle]]s and <math>g,g'</math> are [[Riemannian metric]]s or [[pseudo-Riemannian metric]]s on these). <math>\nabla, \nabla'</math> are [[fact about::metric connection]]s on <math>(E,g)</math> and <math>(E',g')</math> respectively. Then, the [[fact about::tensor product of connections]] <math>\nabla \otimes \nabla'</math> is a metric connection on <math>(E \otimes E',g \otimes g')</math>.<br /> <br /> ==Proof==<br /> <br /> <br /> A [[differential manifold]] <math>M</math>. Two [[fact about::metric bundle]]s <math>(E,g)</math> and <math>(E',g')</math> (i.e., <math>E,E'</math> are [[vector bundle]]s and <math>g,g'</math> are [[Riemannian metric]]s or [[pseudo-Riemannian metric]]s on these). <math>\nabla, \nabla'</math> are [[fact about::metric connection]]s on <math>(E,g)</math> and <math>(E',g')</math> respectively. Then, the [[fact about::tensor product of connections]] <math>\nabla \otimes \nabla'</math> is a metric connection on <math>(E \otimes E',g \otimes g')</math>.<br /> <br /> <br /> Let <math>X,Y \in \Ga(E), X',Y' \in \Ga(E')</math>. Then $\til X \some{def} X \otimes X' \in \Ga(E \otimes E'), \til Y \some{def} Y \otimes Y' \in \Ga(E \otimes E')$. Let $Z \in \Ga(\TM)$. We just expand the two expressions which are needed to be shown equal:<br /> <br /> ==Related facts==<br /> <br /> * [[Tensor product of flat connections is flat]]</div>

Asafs https://diffgeom.subwiki.org/w/index.php?title=Tensor_product_of_metric_connections_is_metric&diff=2111 Tensor product of metric connections is metric 2015-09-22T10:03:09Z

<p>Asafs: /* Proof */</p> <hr /> <div>==Statement==<br /> <br /> A [[differential manifold]] <math>M</math>. Two [[fact about::metric bundle]]s <math>(E,g)</math> and <math>(E',g')</math> (i.e., <math>E,E'</math> are [[vector bundle]]s and <math>g,g'</math> are [[Riemannian metric]]s or [[pseudo-Riemannian metric]]s on these). <math>\nabla, \nabla'</math> are [[fact about::metric connection]]s on <math>(E,g)</math> and <math>(E',g')</math> respectively. Then, the [[fact about::tensor product of connections]] <math>\nabla \otimes \nabla'</math> is a metric connection on <math>(E \otimes E',g \otimes g')</math>.<br /> <br /> ==Proof==<br /> <br /> Let $X,Y \in \Ga(E), X',Y' \in \Ga(E')$. Then $\til X \some{def} X \otimes X' \in \Ga(E \otimes E'), \til Y \some{def} Y \otimes Y' \in \Ga(E \otimes E')$. Let $Z \in \Ga(\TM)$. We just expand the two expressions which are needed to be shown equal:<br /> <br /> ==Related facts==<br /> <br /> * [[Tensor product of flat connections is flat]]</div>

Asafs https://diffgeom.subwiki.org/w/index.php?title=Tensor_product_of_metric_connections_is_metric&diff=2110 Tensor product of metric connections is metric 2015-09-22T10:00:47Z

<p>Asafs: </p> <hr /> <div>==Statement==<br /> <br /> A [[differential manifold]] <math>M</math>. Two [[fact about::metric bundle]]s <math>(E,g)</math> and <math>(E',g')</math> (i.e., <math>E,E'</math> are [[vector bundle]]s and <math>g,g'</math> are [[Riemannian metric]]s or [[pseudo-Riemannian metric]]s on these). <math>\nabla, \nabla'</math> are [[fact about::metric connection]]s on <math>(E,g)</math> and <math>(E',g')</math> respectively. Then, the [[fact about::tensor product of connections]] <math>\nabla \otimes \nabla'</math> is a metric connection on <math>(E \otimes E',g \otimes g')</math>.<br /> <br /> ==Proof==<br /> <br /> <br /> <br /> ==Related facts==<br /> <br /> * [[Tensor product of flat connections is flat]]</div>

Asafs