Close points on a Lie Group with Left-Invariant Metric











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






















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















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






















  • 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










asked yesterday









Duohead

30318




30318












  • 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















active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














 

draft saved


draft discarded


















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






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















 

draft saved


draft discarded



















































 


draft saved


draft discarded














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





















































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







Popular posts from this blog

Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

Image
28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
Read more

ts Property 'filter' does not exist on type '{}'

Image
1 I'm trying to follow Angular Material guide to create a custom filter for autocomplete input text https://material.angular.io/components/autocomplete/overview This is what I have TS import { Component, OnInit } from '@angular/core'; import { TemplateRef } from '@angular/core'; import { FormControl } from "@angular/forms"; import { Observable } from "rxjs"; import { map, startWith, filter } from 'rxjs/operators'; (...) export class AppComponent implements OnInit { title = 'Your Profile'; displayedColumns: string = ['selected', 'name', 'description', 'pnr']; dataSource = ELEMENT_DATA; myControl = new FormControl(); options: any = ['One', 'Two', 'Three']; filteredOptions:
Read more

mat-slide-toggle shouldn't change it's state when I click cancel in confirmation window

Image
1 When I checked the slide toggle it should work as expected. But when I try to uncheck it there is a confirmation window asking if you are sure. When I click cancel in that confirmation window, still the toggle changes to unchecked, which shouldn't happen. It should stay in the same state. Here is my Html and TS Code // html <mat-slide-toggle (change)="change($event)" [checked]="isChecked()" > To-pay </mat-slide-toggle> TS code: // ts change(e) { if(this.checked) { if(confirm("Are you sure")) { console.log("toggle") } else { e.stopPropagation(); console.log("toggle should not change if I click the cancel button") }
Read more