Timeline for Is the Eguchi-Hanson metric a metric on $\mathbb{R}\times S^3$ or tangent bundle of $S^2$?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 16, 2017 at 3:16 | comment | added | mathmetricgeometry | @macbeth: I have multiply the conformal factor, then I get $sec(\partial_r, e_3)=\frac{12r^2}{(r^4-a^4)^2}$. As I listed in the question, the curvature should be $\pm 2,\pm 4$ times $\frac{a^4}{r^6}$. So I don't know where I am wrong. | |
| Nov 15, 2017 at 23:48 | comment | added | macbeth | You were using $ds^2=f^2(r)dr^2+r^2(\sigma_x^2+\sigma_y^2)+r^2 g^2(r)\sigma_z^2$ for the metric before, $dt^2=dr^2+\frac{r^2}{f^2}(\sigma_x^2+\sigma_y^2)+\frac{r^2g^2}{f^2}\sigma_z^2$ now, they differ by the conformal factor $f^2(r)$ which blows up at the zero section! | |
| Nov 15, 2017 at 10:44 | comment | added | mathmetricgeometry | @macbeth: So at the zero section, the sectional curvature is one of $\pm 2, \pm 4$ times $\frac{a^2}{a^6}=\frac{1}{a^4}$? Compute curvature by the the formula for warped products in Peter Petersen's book "Riemannian Geometry". Let $\partial_r$ be the radial direction, $e_1,e_2,e_3$ be the tangent directions. Consider$$dt^2=dr^2+\frac{r^2}{f^2}(\sigma_x^2+\sigma_y^2)+\frac{r^2g^2}{f^2}\sigma_z^2=dr^2+\varphi_1^2(\sigma_x^2+\sigma_y^2)+\varphi_2^2\sigma_z^2$$. Then $sec(\partial_r, e_3)=\frac{1}{f^2(r)} [-\frac{\varphi_2''}{\varphi_2}]=\frac{12r^2}{(r^4-a^4)^2}$, so where do I make a mistake? | |
| Nov 15, 2017 at 4:06 | comment | added | mathmetricgeometry | So at the zero section, the sectional curvature is one of $\pm 2, \pm 4$ times $\frac{a^2}{a^6}=\frac{1}{a^4}$? I compute curvature by formula for warped product in Peter Petersen's book "Riemannian Geometry". Let $\partial_r$ be the radial direction, $e_1,e_2,e_3$ be the tangent directions. Consider the warped product $$dt^2=dr^2+\frac{r^2}{f^2}(\sigma_x^2+\sigma_y^2)+\frac{r^2g^2}{f^2}\sigma_z^2=dr^2+\varphi_1^2(\sigma_x^2+\sigma_y^2)+\varphi_2^2\sigma_z^2.$$ Then $sec(\partial_r, e_3)=\frac{1}{f^2(r)} [-\frac{\varphi_2''}{\varphi_2}]=\frac{12r^2}{(r^4-a^4)^2}$, so where do I make a mistake? | |
| Nov 10, 2017 at 15:22 | comment | added | macbeth | It's a metric on the tangent bundle of $S^2$. The manifold $TS^2$ has a dense open subset (the complement of the zero section) which is diffeomorphic to $(S^3\times \mathbb{R}^+)/\mathbb{Z}_2\cong (S^3/\mathbb{Z}_2)\times (a,\infty)$, so a metric on $TS^2$ can be specified by giving a metric with certain boundary conditions on $S^3\times (a,\infty)$. You should look closely at the construction of the change of variable which "eliminates the singularity". | |
| Nov 10, 2017 at 2:29 | history | edited | mathmetricgeometry | CC BY-SA 3.0 |
edited body
|
| Nov 9, 2017 at 14:37 | history | asked | mathmetricgeometry | CC BY-SA 3.0 |