Are there topologically trivial bundles with a nonzero curvature?
$begingroup$
A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder.
Is it possible to have a nonzero curvature on a topologically trivial fiber bundle? (And if yes, is there any way to visualize this?)
vector-bundles fiber-bundles
asked Jan 14 at 16:43
JakobHJakobH
518318
$endgroup$
add a comment |
$begingroup$
A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder.
Is it possible to have a nonzero curvature on a topologically trivial fiber bundle? (And if yes, is there any way to visualize this?)
vector-bundles fiber-bundles
asked Jan 14 at 16:43
JakobHJakobH
518318
$endgroup$
add a comment |
$begingroup$
A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder.
Is it possible to have a nonzero curvature on a topologically trivial fiber bundle? (And if yes, is there any way to visualize this?)
vector-bundles fiber-bundles
asked Jan 14 at 16:43
JakobHJakobH
518318
$endgroup$
A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder.
Is it possible to have a nonzero curvature on a topologically trivial fiber bundle? (And if yes, is there any way to visualize this?)
vector-bundles fiber-bundles
vector-bundles fiber-bundles
asked Jan 14 at 16:43
JakobHJakobH
518318
asked Jan 14 at 16:43
JakobHJakobH
518318
asked Jan 14 at 16:43
JakobHJakobH
518318
asked Jan 14 at 16:43
JakobHJakobH
518318
asked Jan 14 at 16:43
JakobHJakobH
518318
518318
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $Prightarrow M$ be a non flat bundle, let $U$ be a contractible open subset of $M$, the restriction of $P$ to $U$ is trivial, but the curvature is not always zero. To have a visual picture of this, note that the curvature defines the Lie algebra of the holonomy of the connection (Ambrose Singer) around small loops that you can be supposed contained in $U$.
A concrete example will be the tangent bundle of the sphere and its associated frame bundle, that you restrict to an hemisphere.
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
$endgroup$
add a comment |
Your Answer
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',
autoActivateHeartbeat: false,
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
});
}
});
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%2f3073437%2fare-there-topologically-trivial-bundles-with-a-nonzero-curvature%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $Prightarrow M$ be a non flat bundle, let $U$ be a contractible open subset of $M$, the restriction of $P$ to $U$ is trivial, but the curvature is not always zero. To have a visual picture of this, note that the curvature defines the Lie algebra of the holonomy of the connection (Ambrose Singer) around small loops that you can be supposed contained in $U$.
A concrete example will be the tangent bundle of the sphere and its associated frame bundle, that you restrict to an hemisphere.
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
$endgroup$
add a comment |
$begingroup$
Let $Prightarrow M$ be a non flat bundle, let $U$ be a contractible open subset of $M$, the restriction of $P$ to $U$ is trivial, but the curvature is not always zero. To have a visual picture of this, note that the curvature defines the Lie algebra of the holonomy of the connection (Ambrose Singer) around small loops that you can be supposed contained in $U$.
A concrete example will be the tangent bundle of the sphere and its associated frame bundle, that you restrict to an hemisphere.
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
$endgroup$
add a comment |
$begingroup$
Let $Prightarrow M$ be a non flat bundle, let $U$ be a contractible open subset of $M$, the restriction of $P$ to $U$ is trivial, but the curvature is not always zero. To have a visual picture of this, note that the curvature defines the Lie algebra of the holonomy of the connection (Ambrose Singer) around small loops that you can be supposed contained in $U$.
A concrete example will be the tangent bundle of the sphere and its associated frame bundle, that you restrict to an hemisphere.
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
$endgroup$
Let $Prightarrow M$ be a non flat bundle, let $U$ be a contractible open subset of $M$, the restriction of $P$ to $U$ is trivial, but the curvature is not always zero. To have a visual picture of this, note that the curvature defines the Lie algebra of the holonomy of the connection (Ambrose Singer) around small loops that you can be supposed contained in $U$.
A concrete example will be the tangent bundle of the sphere and its associated frame bundle, that you restrict to an hemisphere.
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
answered Jan 14 at 17:09
Tsemo AristideTsemo Aristide
60.9k11446
60.9k11446
add a comment |
add a comment |
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.
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%2f3073437%2fare-there-topologically-trivial-bundles-with-a-nonzero-curvature%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
