Orientation on normal bundle
$begingroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2832925
$endgroup$
add a comment |
$begingroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2832925
$endgroup$
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
add a comment |
$begingroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2832925
$endgroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2832925
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2832925
edited Jan 2 at 10:52
Bernard
122k740116
edited Jan 2 at 10:52
Bernard
122k740116
edited Jan 2 at 10:52
Bernard
122k740116
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2832925
asked Jan 2 at 10:48
CL.CL.
2,2832925
asked Jan 2 at 10:48
CL.CL.
2,2832925
2,2832925
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
add a comment |
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
$endgroup$
add a comment |
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',
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%2f3059335%2forientation-on-normal-bundle%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
$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
$endgroup$
add a comment |
$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
$endgroup$
add a comment |
$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
$endgroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
59k11445
59k11445
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%2f3059335%2forientation-on-normal-bundle%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
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01