Close points on a Lie Group with Left-Invariant Metric
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
add a comment |
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
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
Nov 21 at 13:44
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 Nov 21 at 13:44
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
Nov 21 at 13:44
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
Nov 21 at 13:44
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
Nov 21 at 13:44
Sorry, I meant Riemannian metric. I'll edit that.
– Duohead
Nov 21 at 13:44
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005126%2fclose-points-on-a-lie-group-with-left-invariant-metric%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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
Nov 21 at 13:44