Examples of odd-dimensional manifolds that do not admit contact structurethe existence of (almost) contact (metric) structureTight vs. overtwisted contact structureThree-dimensional compact Kähler manifoldsIs there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?Is there a “unique” homogeneous contact structure on odd-dimensional spheres?Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds'Unitary' charts on odd-dimensional spheresContact manifolds and pseudodifferential operatorsWhich manifolds admit free involutions?First Chern Class of Contact Structure which is not Torsion
Examples of odd-dimensional manifolds that do not admit contact structure
the existence of (almost) contact (metric) structureTight vs. overtwisted contact structureThree-dimensional compact Kähler manifoldsIs there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?Is there a “unique” homogeneous contact structure on odd-dimensional spheres?Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds'Unitary' charts on odd-dimensional spheresContact manifolds and pseudodifferential operatorsWhich manifolds admit free involutions?First Chern Class of Contact Structure which is not Torsion
$begingroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
$endgroup$
add a comment |
$begingroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
$endgroup$
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^2n+1$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
3 hours ago
add a comment |
$begingroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
$endgroup$
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
dg.differential-geometry at.algebraic-topology differential-topology contact-geometry
edited 2 hours ago
Piotr Hajlasz
9,46843672
9,46843672
asked 3 hours ago
Warlock of Firetop MountainWarlock of Firetop Mountain
25217
25217
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^2n+1$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
3 hours ago
add a comment |
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^2n+1$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
3 hours ago
1
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^2n+1$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
3 hours ago
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^2n+1$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbbS^2n-4$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
$endgroup$
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: "504"
;
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%2fmathoverflow.net%2fquestions%2f325397%2fexamples-of-odd-dimensional-manifolds-that-do-not-admit-contact-structure%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
$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbbS^2n-4$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
$endgroup$
add a comment |
$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbbS^2n-4$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
$endgroup$
add a comment |
$begingroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbbS^2n-4$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
$endgroup$
Every compact orientable $3$-dimensional manifold has a contact structure [1]. On the other hand we have
Theorem. For $ngeq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)timesmathbbS^2n-4$ has no contact structure, see Proposition 2.4 in [2].
[1] J. Martinet,
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[2] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
answered 3 hours ago
Piotr HajlaszPiotr Hajlasz
9,46843672
9,46843672
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f325397%2fexamples-of-odd-dimensional-manifolds-that-do-not-admit-contact-structure%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
1
$begingroup$
Not exactly answering the question since I'm not providing examples, but for a manifold $M^2n+1$, it turns out that admitting a contact structure is equivalent to admitting a reduction of structure group to $U(n) times 1$ (such a reduction is called an almost contact structure), as proved by Borman, Eliashberg, and Murphy. The contact structures they produce for a given almost contact class are overtwisted, meaning they contain some model overtwisted chart. It is a more difficult question to ask when a manifold admits a tight (= non-overtwisted) contact structure.
$endgroup$
– KSackel
3 hours ago