Close points on a Lie Group with Left-Invariant Metric
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Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant Riemannian metric. If $a in G$ is close to the identity $d(a,e) < epsilon$, is $ab$ close to $b$ in general? For $b$ fixed, $a_n to a$ implies that $a_nb to ab$ by continuity. But is there something stonger like $d(ab,b) < C_bepsilon$ for some constant $C_b$ that maybe depends on $b$ in some explicit way?
lie-groups riemannian-geometry
asked Nov 19 at 16:11
Duohead
30818
add a comment |
up vote
1
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Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant Riemannian metric. If $a in G$ is close to the identity $d(a,e) < epsilon$, is $ab$ close to $b$ in general? For $b$ fixed, $a_n to a$ implies that $a_nb to ab$ by continuity. But is there something stonger like $d(ab,b) < C_bepsilon$ for some constant $C_b$ that maybe depends on $b$ in some explicit way?
lie-groups riemannian-geometry
asked Nov 19 at 16:11
Duohead
30818
What is an inner product on a Lie group?
– José Carlos Santos
Nov 19 at 16:15
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
2 days ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant Riemannian metric. If $a in G$ is close to the identity $d(a,e) < epsilon$, is $ab$ close to $b$ in general? For $b$ fixed, $a_n to a$ implies that $a_nb to ab$ by continuity. But is there something stonger like $d(ab,b) < C_bepsilon$ for some constant $C_b$ that maybe depends on $b$ in some explicit way?
lie-groups riemannian-geometry
asked Nov 19 at 16:11
Duohead
30818
Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant Riemannian metric. If $a in G$ is close to the identity $d(a,e) < epsilon$, is $ab$ close to $b$ in general? For $b$ fixed, $a_n to a$ implies that $a_nb to ab$ by continuity. But is there something stonger like $d(ab,b) < C_bepsilon$ for some constant $C_b$ that maybe depends on $b$ in some explicit way?
lie-groups riemannian-geometry
lie-groups riemannian-geometry
asked Nov 19 at 16:11
Duohead
30818
asked Nov 19 at 16:11
Duohead
30818
edited 2 days ago
asked Nov 19 at 16:11
Duohead
30818
asked Nov 19 at 16:11
Duohead
30818
asked Nov 19 at 16:11
Duohead
30818
30818
What is an inner product on a Lie group?
– José Carlos Santos
Nov 19 at 16:15
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
2 days ago
add a comment |
What is an inner product on a Lie group?
– José Carlos Santos
Nov 19 at 16:15
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
2 days ago
What is an inner product on a Lie group?
– José Carlos Santos
Nov 19 at 16:15
What is an inner product on a Lie group?
– José Carlos Santos
Nov 19 at 16:15
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
2 days ago
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
2 days ago
add a comment |
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What is an inner product on a Lie group?
– José Carlos Santos
Nov 19 at 16:15
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
2 days ago