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visits | member for | 2 years, 5 months |
seen | yesterday | |
stats | profile views | 90 |
Feb 26 |
comment |
Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold
@ThiKu your point is well-taken. However I wonder if there is a terminology issue here. There are apparently two kinds of "Killing vectors", those with complete flows and those without. The ones with complete flows are special because they are elements of the Lie algebra of the isometry group. Are both types of vectors commonly referred to as "Killing vectors"? |
Feb 26 |
comment |
Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold
Thank you. This answer together with the comments above have greatly cleared up a lot of my confusion. But how can we tell which, if any, of the pulled-back killing vectors are genuine Killing vectors on $M^m$? For instance, suppose $m=2$, $k=1$, and $dev(M^m)$ is the domain of the stereographic projection $S^2\rightarrow\mathbb{R}^2$. The killing vector that fixes the north pole will pull back to a genuine killing vector, but the others would not. |
Feb 26 |
asked | Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold |
Sep 7 |
awarded | Curious |
Sep 6 |
asked | Casimirs of Poisson brackets obtained via Poisson reduction |
Apr 17 |
accepted | Zero currents localized along a submanifold |
Apr 17 |
comment |
Zero currents localized along a submanifold
The hitch in this definition of $\mathcal{L}_Xs$ is that $s(\phi^X_t(m))$ and $s(m)$ live in different fibers of $E$, and so can't be subtracted. However, when $E$ is a bundle of tensors (and $\mathcal{L}_X$ is the vanilla Lie derivative), I now understand that your ansatz for $u$ is indeed the most general $u$ localized along $S$. Thank you again! |
Apr 17 |
comment |
Zero currents localized along a submanifold
Thanks very much! I have a couple of questions about your response. (i) I'm not sure how those Lie derivatives act on sections of the general vector bundle $E$; if $E$ is not a bundle of tensors, should these be covariant derivatives relative to some connection on E? (ii) If we don't introduce a metric on $M$, do you think the global characterization can be expressed in terms of sections of the abstract normal bundle $(\iota_S^*TM)/TS$? |
Apr 16 |
asked | Zero currents localized along a submanifold |
Jan 17 |
revised |
Is the time derivative of the WKB phase globally defined?
fixed minor error; forgot a factor of 2pi |
Jan 17 |
asked | Is the time derivative of the WKB phase globally defined? |
Nov 5 |
comment |
non-zero, divergence-free vector fields on 2-torus
Thanks very much for your answer. It's pretty slow going for me as I now look through the literature in order to find a more detailed description of why this is true. Could you possibly point me to a reference, maybe a textbook? |
Nov 5 |
awarded | Scholar |
Nov 5 |
awarded | Supporter |
Nov 4 |
asked | non-zero, divergence-free vector fields on 2-torus |
Oct 26 |
comment |
Curl operators parameterized by the set of Riemannien metrics on a 3-manifold
@Paul Reynolds : Sorry for not being clear. I'm not requiring the volume form to be the one determined by the metric. |
Oct 25 |
revised |
non-vanishing magnetic helicty density
another example |
Oct 25 |
revised |
Curl operators parameterized by the set of Riemannien metrics on a 3-manifold
hopefully clarified question |
Oct 25 |
asked | Curl operators parameterized by the set of Riemannien metrics on a 3-manifold |
Oct 19 |
revised |
non-vanishing magnetic helicty density
slight generalization of question |