Timeline for Building a geodesic conjugate parameterization on catenoid
Current License: CC BY-SA 4.0
9 events
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| Jul 20, 2023 at 17:20 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed some typos and corrected a few formulae
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| Jul 17, 2023 at 12:58 | comment | added | Robert Bryant | @ArvinRasoulzadeh: Yes, it follows from this: If $g ={ \omega_1}^2 + {\omega_2}^2$ where $\omega_1 = Z^\flat$, and $\kappa(p)$ is the geodesic curvature of the flow line of $Z$ through a point $p$ of the surface, then $\mathrm{d}\omega_1 = \kappa\,\omega_1\wedge\omega_2$ (a consequence of the structure equations). Thus, $\omega_1$ is closed if and only if $\kappa$ vanishes identically, i.e., the flow lines of $Z$ are geodesics. | |
| Jul 17, 2023 at 12:18 | comment | added | RWien | I got really interested in this sentence: "the condition that the flow lines of 𝑍 be geodesics is that 𝑍♭, the dual 1-form, be closed." I know that this might be a trivial question but can you please tell me how we know this? | |
| Jul 16, 2023 at 9:56 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a remark about a case where there do exist solutions.
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| Jul 14, 2023 at 16:53 | vote | accept | RWien | ||
| Jul 14, 2023 at 14:34 | comment | added | Robert Bryant | @ArvinRasoulzadeh: They are not the coordinate vector fields, but they are multiples of the coordinate vector fields. You can always take the coordinate vector fields (which are nonvanishing) and divide by their lengths in the induced metric. The resulting unit vector fields $X$ and $Y$ aren't coordinate vector fields any more because $[X,Y]$ is (usually) nonzero, but that doesn't affect whether they are conjugate with respect to $I\!I$. | |
| Jul 14, 2023 at 14:01 | comment | added | RWien | How can we assume that the coordinate vector field $X$ and $Y$ are unit vector fields? This is not obvious for me or maybe I'm missing something. | |
| Jul 14, 2023 at 13:03 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Rewrote the argument for clarity
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| Jul 12, 2023 at 20:24 | history | answered | Robert Bryant | CC BY-SA 4.0 |