Mixing
Relationship between Carnot-Caratheory Distance and Levi-Civita Connection
$begingroup$
Suppose that $Gcong Htimes K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that
$H$ is Commutative
$K$ is Compact
Then $G$ admits a bi-invariant metric then the (Levi-Civita) Riemannian exponential and Lie Group exponential maps coincide. Since the Riemannian structure induces a natural distance structure, then the metric can be expressed using the Lie group structure also.
However, since $G$ is nilpotent then the Carnot-Caratheory distance also is well-defined.
Question
My question, is how are these two structures related? Since they both seem to be unique and induced by $G$'s nice Lie Group geometry.
lie-groups riemannian-geometry connections nilpotent-groups
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
$endgroup$
add a comment |
$begingroup$
Suppose that $Gcong Htimes K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that
$H$ is Commutative
$K$ is Compact
Then $G$ admits a bi-invariant metric then the (Levi-Civita) Riemannian exponential and Lie Group exponential maps coincide. Since the Riemannian structure induces a natural distance structure, then the metric can be expressed using the Lie group structure also.
However, since $G$ is nilpotent then the Carnot-Caratheory distance also is well-defined.
Question
My question, is how are these two structures related? Since they both seem to be unique and induced by $G$'s nice Lie Group geometry.
lie-groups riemannian-geometry connections nilpotent-groups
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
$endgroup$
add a comment |
$begingroup$
Suppose that $Gcong Htimes K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that
$H$ is Commutative
$K$ is Compact
Then $G$ admits a bi-invariant metric then the (Levi-Civita) Riemannian exponential and Lie Group exponential maps coincide. Since the Riemannian structure induces a natural distance structure, then the metric can be expressed using the Lie group structure also.
However, since $G$ is nilpotent then the Carnot-Caratheory distance also is well-defined.
Question
My question, is how are these two structures related? Since they both seem to be unique and induced by $G$'s nice Lie Group geometry.
lie-groups riemannian-geometry connections nilpotent-groups
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
$endgroup$
Suppose that $Gcong Htimes K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that
$H$ is Commutative
$K$ is Compact
Then $G$ admits a bi-invariant metric then the (Levi-Civita) Riemannian exponential and Lie Group exponential maps coincide. Since the Riemannian structure induces a natural distance structure, then the metric can be expressed using the Lie group structure also.
However, since $G$ is nilpotent then the Carnot-Caratheory distance also is well-defined.
Question
My question, is how are these two structures related? Since they both seem to be unique and induced by $G$'s nice Lie Group geometry.
lie-groups riemannian-geometry connections nilpotent-groups
lie-groups riemannian-geometry connections nilpotent-groups
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
asked Jan 28 at 15:25
AIM_BLBAIM_BLB
2,5342820
2,5342820
add a comment |
add a comment |
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