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We assume positivity for the function \Omega, since it could be zero at some point and then we have no metric.
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edit approved Dec 6, 2021 at 16:45
Julian Seipel
edit approved Dec 6, 2021 at 16:45
Julian Seipel
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Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric on $S^2$)? If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric on $S^2$)? If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric on $S^2$)? If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.

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LSpice
LSpice
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Are metrics of the form $dr^2+ \Omega^2 r^2 g_g_\text{round}$ asymptotically flat?

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_{round}$$dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat? (where $g_{round}$$g_\text{round}$ is the round metric on $S^2$.)? If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.

Are metrics of the form $dr^2+ \Omega^2 r^2 g_{round}$ asymptotically flat?

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_{round}$ asymptotically flat? (where $g_{round}$ is the round metric on $S^2$.) If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.

Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric on $S^2$)? If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.

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Laithy
Laithy
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Are metrics of the form $dr^2+ \Omega^2 r^2 g_{round}$ asymptotically flat?

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_{round}$ asymptotically flat? (where $g_{round}$ is the round metric on $S^2$.) If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.