Timeline for Algebraic geometric conditions on the variety $V(F)$ such that the manifold defined by $F$ has nonvanishing second fundamental form?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Nov 24, 2020 at 16:41 | history | bounty ended | Johnny T. | ||
| S Nov 24, 2020 at 16:41 | history | notice removed | Johnny T. | ||
| S Nov 22, 2020 at 18:40 | history | bounty started | Johnny T. | ||
| S Nov 22, 2020 at 18:40 | history | notice added | Johnny T. | Reward existing answer | |
| Nov 22, 2020 at 18:39 | vote | accept | Johnny T. | ||
| Nov 21, 2020 at 21:39 | answer | added | Robert Bryant | timeline score: 1 | |
| Nov 20, 2020 at 13:11 | comment | added | Johnny T. | @BenMcKay Thank you for your comments! | |
| Nov 20, 2020 at 10:18 | comment | added | Ben McKay | The argument for the quadric is very special: there are very few homogeneous projective hypersurfaces. | |
| Nov 20, 2020 at 10:16 | comment | added | Johnny T. | @BenMcKay Thank you for the clarification. I suppose the argument does not generalise to higher degree? | |
| Nov 19, 2020 at 17:41 | comment | added | Ben McKay | yes, the smooth quadric hypersurface is invariant under a group of symmetries which acts transitively on the real points of its cone, i.e. any two nonzero null vectors of a real quadratic form are carried to one another by a linear transformation preserving the quadratic form. The second fundamental form is a projective invariant, so its null space is too. The second fundamental form can't vanish everywhere, because the quadric is not linear. | |
| Nov 19, 2020 at 17:21 | comment | added | Johnny T. | @BenMcKay Did I understand your last sentence correctly?: if the degree of $F$ is $2$ in the question then $M$ has nowhere vanishing second fundamental form (meaning as in the Edit)? | |
| Nov 19, 2020 at 16:07 | comment | added | Ben McKay | Thanks, that's clear. | |
| Nov 19, 2020 at 14:00 | comment | added | Johnny T. | I have added an edit. Please let me know if the question still contains ambiguity. Thank you! | |
| Nov 19, 2020 at 12:57 | history | edited | Johnny T. | CC BY-SA 4.0 |
added 158 characters in body
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| Nov 19, 2020 at 12:28 | comment | added | Ben McKay | Your $M$ is a cone, so contains lines (through the origin), and these are geodesics in the ambient space, so the second fundamental form vanishes on them. So you might clarify, do you mean that the second fundamental form is nonzero on some tangent vector, or on every nonzero tangent vector? A smooth quadric hypersurface clearly has homogeneous cone, so no tangent space has zero second fundamental form. | |
| Nov 19, 2020 at 9:06 | history | asked | Johnny T. | CC BY-SA 4.0 |