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 inner product. 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







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  • What is an inner product on a Lie group?
    – José Carlos Santos
    yesterday















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Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant inner product. 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







share|cite|improve this question






















  • What is an inner product on a Lie group?
    – José Carlos Santos
    yesterday













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant inner product. 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







share|cite|improve this question













Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant inner product. 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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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  • What is an inner product on a Lie group?
    – José Carlos Santos
    yesterday


















  • What is an inner product on a Lie group?
    – José Carlos Santos
    yesterday
















What is an inner product on a Lie group?
– José Carlos Santos
yesterday




What is an inner product on a Lie group?
– José Carlos Santos
yesterday















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