Mixing
Adapted connection on foliation
$begingroup$
Let $(M,F)$ be a manifold with foliation, fixing a bundle-like metric $g$ we identifying $Q=TM/Fcong F^perp$, we define the connection on $Q$, for any $sinGamma(Q)$
$$nabla_Xs=begin{cases}
([X,s])^perp,& Xin Gamma(F),\
(nabla^g_Xs)^perp,& Xin Gamma(F^perp).
end{cases}$$
Q How to show that $nabla$ is holonomy invaraint, i.e.
$$L_V(nabla)(Y,s)=0?$$
riemannian-geometry foliations
asked Feb 1 at 3:30
DLINDLIN
406414
$endgroup$
add a comment |
$begingroup$
Let $(M,F)$ be a manifold with foliation, fixing a bundle-like metric $g$ we identifying $Q=TM/Fcong F^perp$, we define the connection on $Q$, for any $sinGamma(Q)$
$$nabla_Xs=begin{cases}
([X,s])^perp,& Xin Gamma(F),\
(nabla^g_Xs)^perp,& Xin Gamma(F^perp).
end{cases}$$
Q How to show that $nabla$ is holonomy invaraint, i.e.
$$L_V(nabla)(Y,s)=0?$$
riemannian-geometry foliations
asked Feb 1 at 3:30
DLINDLIN
406414
$endgroup$
add a comment |
$begingroup$
Let $(M,F)$ be a manifold with foliation, fixing a bundle-like metric $g$ we identifying $Q=TM/Fcong F^perp$, we define the connection on $Q$, for any $sinGamma(Q)$
$$nabla_Xs=begin{cases}
([X,s])^perp,& Xin Gamma(F),\
(nabla^g_Xs)^perp,& Xin Gamma(F^perp).
end{cases}$$
Q How to show that $nabla$ is holonomy invaraint, i.e.
$$L_V(nabla)(Y,s)=0?$$
riemannian-geometry foliations
asked Feb 1 at 3:30
DLINDLIN
406414
$endgroup$
Let $(M,F)$ be a manifold with foliation, fixing a bundle-like metric $g$ we identifying $Q=TM/Fcong F^perp$, we define the connection on $Q$, for any $sinGamma(Q)$
$$nabla_Xs=begin{cases}
([X,s])^perp,& Xin Gamma(F),\
(nabla^g_Xs)^perp,& Xin Gamma(F^perp).
end{cases}$$
Q How to show that $nabla$ is holonomy invaraint, i.e.
$$L_V(nabla)(Y,s)=0?$$
riemannian-geometry foliations
riemannian-geometry foliations
asked Feb 1 at 3:30
DLINDLIN
406414
asked Feb 1 at 3:30
DLINDLIN
406414
edited Feb 1 at 3:40
DLIN
asked Feb 1 at 3:30
DLINDLIN
406414
asked Feb 1 at 3:30
DLINDLIN
406414
asked Feb 1 at 3:30
DLINDLIN
406414
406414
add a comment |
add a comment |
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