Einstein metrics on spheresSmooth Poincaré Conjecturerescaled metric quantities on rescaling...
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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 4 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 4 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 4 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 4 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
edited 4 hours ago
Michael Albanese
64.6k1599315
asked 4 hours ago
Bruce WayneBruce Wayne
448213
edited 4 hours ago
Michael Albanese
64.6k1599315
edited 4 hours ago
Michael Albanese
64.6k1599315
edited 4 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 4 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
3 hours ago
add a comment |
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1 Answer
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votes
1 Answer
1
active
oldest
votes
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oldest
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$begingroup$
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 4 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
3 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 4 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
3 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 4 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 4 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
edited 3 hours ago
answered 4 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
answered 4 hours ago
Michael AlbaneseMichael Albanese
64.6k1599315
answered 4 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
3 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
3 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
3 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
3 hours ago
add a comment |
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