Einstein metrics on spheresSmooth Poincaré Conjecturerescaled metric quantities on rescaling...
Mapping arrows in commutative diagrams
extract characters between two commas?
How to answer pointed "are you quitting" questioning when I don't want them to suspect
Shall I use personal or official e-mail account when registering to external websites for work purpose?
Why is making salt water prohibited on Shabbat?
Travelling to Edinburgh from India
Why airport relocation isn't done gradually?
Can produce flame be used to grapple, or as an unarmed strike, in the right circumstances?
Some basic questions on halt and move in Turing machines
aging parents with no investments
Piano - What is the notation for a double stop where both notes in the double stop are different lengths?
Why is my log file so massive? 22gb. I am running log backups
Could a US political party gain complete control over the government by removing checks & balances?
Why did the Germans forbid the possession of pet pigeons in Rostov-on-Don in 1941?
Why is the design of haulage companies so “special”?
Why was the "bread communication" in the arena of Catching Fire left out in the movie?
How can I add custom success page
Add an angle to a sphere
Why do UK politicians seemingly ignore opinion polls on Brexit?
Manga about a female worker who got dragged into another world together with this high school girl and she was just told she's not needed anymore
How to manage monthly salary
Email Account under attack (really) - anything I can do?
OA final episode explanation
Is there a way to make member function NOT callable from constructor?
Einstein metrics on spheres
Smooth Poincaré Conjecturerescaled metric quantities on rescaling metricsReversing the Ricci flowThoughts about sectional curvatureEasy solution to Yamabe problem for surfacesShowing that Ricci curvature of round unit sphere $(S^n,g_0)$ is $Ric(g_0)=(n-1)g_0$Uniformization of metrics vs. uniformization of Riemann surfacesCounterexample to Gunther Theorem when assuming only a Ricci curvature upper boundPointwise conformal vs. conformally diffeomorphic metrics in dimension 2Upper volume bounds for submanifolds
$begingroup$
I've got a couple of quick questions that came up after reading a peculiar statement in some article. The sentence says something like "... is the $N$-dimensional sphere with constant Ricci curvature equal to $K$...", and the questions are something like:
For $(mathbb{S}^n,g)$ the sphere with its standard differential structure and $some$ Riemannian metric on it,
1.a. Does $g$ being an Einstein metric implies that it is actually the round metric (up to some normalization constant)?
1.b. Does the answer change if we change to an alternative differential structure (when possible)?
I guess this shouldn't be true, so in this case
2. Is there an intuitive way to understand how one could construct a metric which is Einstein but not of constant curvature?
Anyways, I thank you all in advance for sharing your knowledge.
differential-geometry riemannian-geometry
edited 3 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
$endgroup$
add a comment |
$begingroup$
I've got a couple of quick questions that came up after reading a peculiar statement in some article. The sentence says something like "... is the $N$-dimensional sphere with constant Ricci curvature equal to $K$...", and the questions are something like:
For $(mathbb{S}^n,g)$ the sphere with its standard differential structure and $some$ Riemannian metric on it,
1.a. Does $g$ being an Einstein metric implies that it is actually the round metric (up to some normalization constant)?
1.b. Does the answer change if we change to an alternative differential structure (when possible)?
I guess this shouldn't be true, so in this case
2. Is there an intuitive way to understand how one could construct a metric which is Einstein but not of constant curvature?
Anyways, I thank you all in advance for sharing your knowledge.
differential-geometry riemannian-geometry
edited 3 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
$endgroup$
add a comment |
$begingroup$
I've got a couple of quick questions that came up after reading a peculiar statement in some article. The sentence says something like "... is the $N$-dimensional sphere with constant Ricci curvature equal to $K$...", and the questions are something like:
For $(mathbb{S}^n,g)$ the sphere with its standard differential structure and $some$ Riemannian metric on it,
1.a. Does $g$ being an Einstein metric implies that it is actually the round metric (up to some normalization constant)?
1.b. Does the answer change if we change to an alternative differential structure (when possible)?
I guess this shouldn't be true, so in this case
2. Is there an intuitive way to understand how one could construct a metric which is Einstein but not of constant curvature?
Anyways, I thank you all in advance for sharing your knowledge.
differential-geometry riemannian-geometry
edited 3 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
$endgroup$
I've got a couple of quick questions that came up after reading a peculiar statement in some article. The sentence says something like "... is the $N$-dimensional sphere with constant Ricci curvature equal to $K$...", and the questions are something like:
For $(mathbb{S}^n,g)$ the sphere with its standard differential structure and $some$ Riemannian metric on it,
1.a. Does $g$ being an Einstein metric implies that it is actually the round metric (up to some normalization constant)?
1.b. Does the answer change if we change to an alternative differential structure (when possible)?
I guess this shouldn't be true, so in this case
2. Is there an intuitive way to understand how one could construct a metric which is Einstein but not of constant curvature?
Anyways, I thank you all in advance for sharing your knowledge.
differential-geometry riemannian-geometry
differential-geometry riemannian-geometry
edited 3 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
edited 3 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
edited 3 hours ago
Michael Albanese
64.6k1599315
edited 3 hours ago
Michael Albanese
64.6k1599315
edited 3 hours ago
Michael Albanese
64.6k1599315
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
asked 4 hours ago
Bruce WayneBruce Wayne
448213
asked 4 hours ago
Bruce WayneBruce Wayne
448213
448213
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, there are Einstein metrics on spheres which are not rescalings of the round metric. See the introduction of Einstein metrics on spheres by Boyer, Galicki, & Kollár for some constructions. However, as far as I am aware, there are no known examples of Einstein metrics with non-positive Einstein constant. In particular, it is an open question as to whether $S^n$ admits a Ricci-flat metric for $n geq 4$.
If we consider exotic spheres, they do not admit a 'round metric' or any metric of constant curvature, so I'm not sure what is meant by this. However, there are examples of Einstein metrics on exotic spheres, see Einstein Metrics on Exotic Spheres in Dimensions 7, 11, and 15 by Boyer, Galicki, Kollár, & Thomas for example. Note however that there are some exotic spheres which, if they admit Einstein metrics, must have negative Einstein constant.
Finding Einstein metrics which are not constant curvature is, in general, a hard thing to do and an area of active research.
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
$endgroup$
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
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%2f3179975%2feinstein-metrics-on-spheres%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
$begingroup$
No, there are Einstein metrics on spheres which are not rescalings of the round metric. See the introduction of Einstein metrics on spheres by Boyer, Galicki, & Kollár for some constructions. However, as far as I am aware, there are no known examples of Einstein metrics with non-positive Einstein constant. In particular, it is an open question as to whether $S^n$ admits a Ricci-flat metric for $n geq 4$.
If we consider exotic spheres, they do not admit a 'round metric' or any metric of constant curvature, so I'm not sure what is meant by this. However, there are examples of Einstein metrics on exotic spheres, see Einstein Metrics on Exotic Spheres in Dimensions 7, 11, and 15 by Boyer, Galicki, Kollár, & Thomas for example. Note however that there are some exotic spheres which, if they admit Einstein metrics, must have negative Einstein constant.
Finding Einstein metrics which are not constant curvature is, in general, a hard thing to do and an area of active research.
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
$endgroup$
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
add a comment |
$begingroup$
No, there are Einstein metrics on spheres which are not rescalings of the round metric. See the introduction of Einstein metrics on spheres by Boyer, Galicki, & Kollár for some constructions. However, as far as I am aware, there are no known examples of Einstein metrics with non-positive Einstein constant. In particular, it is an open question as to whether $S^n$ admits a Ricci-flat metric for $n geq 4$.
If we consider exotic spheres, they do not admit a 'round metric' or any metric of constant curvature, so I'm not sure what is meant by this. However, there are examples of Einstein metrics on exotic spheres, see Einstein Metrics on Exotic Spheres in Dimensions 7, 11, and 15 by Boyer, Galicki, Kollár, & Thomas for example. Note however that there are some exotic spheres which, if they admit Einstein metrics, must have negative Einstein constant.
Finding Einstein metrics which are not constant curvature is, in general, a hard thing to do and an area of active research.
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
$endgroup$
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
add a comment |
$begingroup$
No, there are Einstein metrics on spheres which are not rescalings of the round metric. See the introduction of Einstein metrics on spheres by Boyer, Galicki, & Kollár for some constructions. However, as far as I am aware, there are no known examples of Einstein metrics with non-positive Einstein constant. In particular, it is an open question as to whether $S^n$ admits a Ricci-flat metric for $n geq 4$.
If we consider exotic spheres, they do not admit a 'round metric' or any metric of constant curvature, so I'm not sure what is meant by this. However, there are examples of Einstein metrics on exotic spheres, see Einstein Metrics on Exotic Spheres in Dimensions 7, 11, and 15 by Boyer, Galicki, Kollár, & Thomas for example. Note however that there are some exotic spheres which, if they admit Einstein metrics, must have negative Einstein constant.
Finding Einstein metrics which are not constant curvature is, in general, a hard thing to do and an area of active research.
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
$endgroup$
No, there are Einstein metrics on spheres which are not rescalings of the round metric. See the introduction of Einstein metrics on spheres by Boyer, Galicki, & Kollár for some constructions. However, as far as I am aware, there are no known examples of Einstein metrics with non-positive Einstein constant. In particular, it is an open question as to whether $S^n$ admits a Ricci-flat metric for $n geq 4$.
If we consider exotic spheres, they do not admit a 'round metric' or any metric of constant curvature, so I'm not sure what is meant by this. However, there are examples of Einstein metrics on exotic spheres, see Einstein Metrics on Exotic Spheres in Dimensions 7, 11, and 15 by Boyer, Galicki, Kollár, & Thomas for example. Note however that there are some exotic spheres which, if they admit Einstein metrics, must have negative Einstein constant.
Finding Einstein metrics which are not constant curvature is, in general, a hard thing to do and an area of active research.
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
edited 3 hours ago
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
answered 3 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
64.6k1599315
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
add a comment |
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
$begingroup$
great, thanks! Yeah, of course you are right, question 1b doesn't make sense as stated. I wrote it fast, sorry!
$endgroup$
– Bruce Wayne
2 hours ago
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%2f3179975%2feinstein-metrics-on-spheres%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
