User martin gisser - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:42:04Z http://mathoverflow.net/feeds/user/9161 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54343/is-there-a-preferable-convention-for-defining-the-wedge-product/122631#122631 Answer by Martin Gisser for Is there a preferable convention for defining the wedge product? Martin Gisser 2013-02-22T14:38:34Z 2013-02-22T14:38:34Z <p>I prefer (alas!) the Kobayashi-Nomizu "algebraic" version. Besides the <em>formal benefits</em> of having a projector and not needing to carry around combinatorial factors, here is a <em>differential geometric case</em>:</p> <p>If $\nabla \alpha$ is the Levi-Civita connection applied to the 1-form $\alpha$, then the exterior differential $d\alpha$ is the antisymmetric part of $\nabla\alpha$. In the Spivak "geometer" version it would be $\frac 1 2 d\alpha$.</p> <p>(The symmetric part of this decompostion involves the Lie derivative of the metric. See the nice book by W.A. Poor, Differential Geometric Structures, or Peter Petersen's Riemannian Geometry 2nd ed. (Who has found this wondrous decomposition?))</p> <p><strong>I hope I'm confused on this...</strong> 1) Two different "canonical" exterior differentials would be quite a scandalous mess. 2) I don't know any textbook mentioning this issue</p> <p>P.S.: see appendix of <a href="http://arxiv.org/abs/math-ph/0212043" rel="nofollow">http://arxiv.org/abs/math-ph/0212043</a> showing what bad can happen</p> http://mathoverflow.net/questions/39686/which-journals-publish-1-page-papers Which journals publish 1-page papers Martin Gisser 2010-09-23T04:47:14Z 2012-07-08T16:09:44Z <p>Having read a thread on a similar question on expository papers I'm reminded of reason #99 to drop my math PhD thingy, late c20th: I just couldn't blow up this paper to 4 pages. (OK, one half-page calculation was left to the expert reader (and other experts could guess), but...)</p> http://mathoverflow.net/questions/39540/is-cartans-magic-formula-due-to-lie-or-henri Is "Cartan's magic formula" due to Élie or Henri? Martin Gisser 2010-09-21T21:21:59Z 2012-03-26T13:05:57Z <p>The formula $\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. Berline, Getzler, Vergne; Heat Kernels and Dirac Operators, p.17), and often just to "Cartan" (e.g. <a href="http://en.wikipedia.org/wiki/Lie_derivative" rel="nofollow">http://en.wikipedia.org/wiki/Lie_derivative</a> ). Who is right? Reference?</p> http://mathoverflow.net/questions/74334/helmholtz-decomposition-on-compact-riemannian-manifolds/87887#87887 Answer by Martin Gisser for Helmholtz-Decomposition on compact Riemannian manifolds Martin Gisser 2012-02-08T12:56:04Z 2012-02-08T12:56:04Z <p>Günter Schwarz, <em>Hodge Decomposition - A Method for Solving Boundary Value Problems</em>, Lecture Notes in Maths <strong>1607</strong> (1995)</p> http://mathoverflow.net/questions/86666/existence-uniqueness-and-regularity-for-linear-parabolic-pde-on-a-complete-riem/86866#86866 Answer by Martin Gisser for Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold Martin Gisser 2012-01-27T23:06:28Z 2012-01-27T23:26:37Z <p>I've played with such things 15 years ago. Here's what I remember...</p> <p>Sure you need some semi-boundedness condition on K. Uniqueness should then follow via semigroup domination (aka Kato's inequality) from scalar case.</p> <p>I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108</p> <p>Shigekawa had rediscovered Barry Simon's semigroup domination criterion (i.e. Kato's inequality) and generalized it to the vector valued things.</p> <p>See also:</p> <p>R.S. Strichartz, Analysis of the Laplacian on the Complete Riemannian Manifold, J. Funct. Analysis 52 (1983), 48-79. Scalar case: Theorem 3.5 - but has uniqueness only for 1&lt; p&lt;∞.</p> <p>(Some details in Strichartz' proof(s) for vector valued Laplacians not optimal (classic problems with doing tensor calculus and Stokes...) not even in the follow-up paper "L^p contractive projections and the heat semigroup for differential forms" J. Funct. Analysis, 65 (1985), 348-357)</p> <p>Perhaps look also for El Maati Ouhabaz ca. 1999. He's written a book, "Analysis of Heat Equations on Domains" (2004) but I never got hold of it.</p> http://mathoverflow.net/questions/61270/infinitesimal-generators-of-stochastic-processes/79434#79434 Answer by Martin Gisser for Infinitesimal generators of stochastic processes Martin Gisser 2011-10-28T22:23:17Z 2011-10-28T23:19:08Z <p>The most important thing is ψ≥0 ⇒ U(t)ψ≥0. (The stochastician in me would be content with a substochastic semigroup (Markov processes might die), and the Hilbert spaced part of me would add: well, that's simply the condition of the generator being negative definite.)</p> <p>It seems the definite treatment of ψ≥0 ⇒ U(t)ψ≥0 is theorem 1.6 in</p> <p>Wolfgang Arendt, <a href="http://cantor.mathematik.uni-ulm.de/m5/arendt/publications/arendt-pub/short/1984-Are-KtsInq.pdf" rel="nofollow">Kato's Inequality: A Characterisation of Generators of Positive Semigroups</a>, <em>Proc. R. Ir. Acad.</em> Vol. <strong>84A</strong> No. 2 (1984), 155-174</p> <pre><code> ~ . ~ . ~ </code></pre> <p><strong>(</strong> Arendt also has the <em>semigroup domination</em> theorem 4.3 plus important ramifications. Until yesterday I called it the Kato-Simon-Shigekawa criterion. But no, according to Arendt he learned it from Kato himself. I've studied a bit of this stuff ca. 1995 (told Shigekawa about Simon's proof, and still (2011) have the most elegant verification of Kato's inequalities for general manifolds...) - but <em>never did I come across Arendt's paper!</em> (Well, these studies were laying so dormant in my ol brains that John's question of April 2011 couldn't shake them fully awake. It took a second hit from somewhere else.) Mestupid did quite some digging this week (plus, 3 failed attempts of proof of Arendt's theorem (in $L^1$) plus a total recall of all my higher analysis from gone times) to finally hit the paper... <strong>)</strong></p> <pre><code> ~ . ~ . ~ </code></pre> <p>This comment dedicated in memoriam Johann Schneidermeier</p> http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g/59662#59662 Answer by Martin Gisser for Applications of Koszul's formula other than the fundamental lemma of Riemannian geometry Martin Gisser 2011-03-26T15:02:27Z 2011-03-26T15:02:27Z <p>I've meanwhile found out that my innocent example ... <p><b>1.)</b> ... amounts to a rediscovery of Schouten's <i><a href="http://en.wikipedia.org/wiki/Contorsion_tensor" rel="nofollow">contorsion tensor</a></i>. (See e.g. <a href="http://www.telesio-galilei.com/tg/images/stories/documents/discussion/riemann%20cartan%20connection%20wa%20rodrigues%20jr%2027-11-2008.pdf" rel="nofollow">this note</a>). <p>This concept is important in <a href="http://arxiv.org/abs/gr-qc/0501017" rel="nofollow">torsion gravity</a> (which isn't as exotic as it may sound - except for some <a href="http://en.wikipedia.org/wiki/Torsion_field" rel="nofollow">charlatanry</a> derived from it...) My example equation seems to express the equivalence principle (cf. <a href="http://arxiv.org/abs/gr-qc/0501017" rel="nofollow">link</a> eq. (255)). See also Rodrigues &amp; Oliveira: <a href="http://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-540-71292-3" rel="nofollow">The many faces of Maxwell, Dirac and Einstein equations</a>.</p> <p><p>As I've already hinted here, contorsion stuff (or what I would term the <i>contorsion operator</i>) can also occur in computations with torsion-free connections.</p> <p><p> <b>2.)</b> ... leads to a generalization of the fundamental "lemma", a.k.a. Levi-Civita theorem: <i>For any given vector-valued 2-form $T$ there exists a unique metric connection with torsion $T$</i>.</p> http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g Applications of Koszul's formula other than the fundamental lemma of Riemannian geometry Martin Gisser 2011-03-07T10:06:43Z 2011-03-26T15:02:27Z <p>I'm wondering what else one can do with Koszul's formula</p> <p>$$2\langle\nabla_XA,B\rangle = X\langle A,B\rangle-B\langle X,A\rangle + A\langle X,B\rangle - \langle A,[X,B]\rangle + \langle[B,X],A\rangle - \langle B,[A,X]\rangle$$</p> <p>beyond proving existence and uniqueness of the Levi-Civita connection. I haven't yet seen anybody using it for anything else, which would be quite curious.</p> <p>Here's a pretty and simple example. I don't know if it is known...</p> <p>Let $\nabla^{LC}$ be the Levi-Civita connection and $\nabla$ be some other metric connection with torsion $T$. Then</p> <p>$$2\langle\nabla_XA,B\rangle - 2\langle\nabla_X^{LC}A,B\rangle= \langle T(A,X),B\rangle -\langle T(A,B),X\rangle - \langle T(B,X),A\rangle$$</p> <p>An application of this would be to compute the LC-connection for the metric $\langle K\cdot,K\cdot\rangle$ in terms of the endomorphism $K$ and the LC-connection for $\langle \cdot,\cdot\rangle$. The computation starts with the connection $K^{-1}\nabla^{LC} K$, which is metric for $\langle K\cdot,K\cdot\rangle$...</p> <p>I can't guarantee correct letters and signs :-)</p> http://mathoverflow.net/questions/39676/which-bianchi-identity-is-due-to-bianchi-or-not-since-it-might-be-due-to-ricci Which Bianchi identity is due to Bianchi (or not, since it might be due to Ricci (according to Levi-Civita (according to MO))) or vice versa? Martin Gisser 2010-09-23T02:06:00Z 2010-12-12T08:07:45Z <p>wikipedia doesn't say, nor my Berger Panorama book (but I might google Levi-Civita to get rid of one level of brackets) and the library is far (actually not, but it has German Schließungszeiten and I got no key). (I guess the differential Bianchi identity is not due to Bianchi? So who did which?)</p> <p>To clarify on the "vice versa": according to my Swiss cheese memory, there might be something completely different actually due to Bianchi. Some book was telling this quite nice...</p> <p>Said MO answer is <a href="http://mathoverflow.net/questions/24132/what-are-examples-of-mathematical-concepts-named-after-the-wrong-people-stigler/24154#24154" rel="nofollow">here</a></p> http://mathoverflow.net/questions/14877/how-much-of-differential-geometry-can-be-developed-entirely-without-atlases/38405#38405 Answer by Martin Gisser for How much of differential geometry can be developed entirely without atlases? Martin Gisser 2010-09-11T12:43:05Z 2010-09-23T06:18:54Z <p>I'm on a similar quest and have a tentative answer (I'm currently working it out). My definition of a generalized differential manifold would be: A locally ringed space with nontrivial cotangent sheaf. Covariant derivatives can be defined using only the cotangent sheaf, and there is a dual version of the fundamental lemma of Riemannian geometry for this. (My creed: forget tangent space.)</p> <p>The abstract differential geometry of Mallios et al looks quite attractive to me, but I haven't yet read it in detail (lacking time, money, and a useable math library). One immediate problem I see with their definition of vector bundle, being the usual suspect, a locally finite and free sheaf module. But a major motivation for using sheaves is infinite dimensional applications, like path space (I suspect path space should be introduced before geodesics, Jacobi fields, General Relativity, perhaps Brownian motion... Dunno how sheaves connect to uniform spaces...).</p> <p>I have an incomplete copy of yummy lectures by Jürgen Bingener, Regensburg 1983/4 on basics of calculus and Riemannian geometry in the language of ringed spaces. Need to contact him and ask for more.</p> http://mathoverflow.net/questions/15366/which-journals-publish-expository-work/39684#39684 Answer by Martin Gisser for Which journals publish expository work? Martin Gisser 2010-09-23T04:02:55Z 2010-09-23T04:02:55Z <p>I want to second Deane's comment on the usefulness of blogs (wikis). We can TeX, meanwhile, anywhere, anyhow. There could be a very fruitful symbiosis between blog and journal. The blog symbiont is the garden where expositions grow, perhaps fertilized if not seeded by commenters. Each symbiont consists of two poles: medium vs. reader. The journal symbiont faces the problem that its reader can not always write in it (or the marginals get closed away in one library somewhere). So, blogs (wikis) can also serve the marginals, <i>something the medieval libraries knew to value.</i></p> <p>This might hint at a new internet gadget. Which reminds me of another gadget: The user-sorted printout shelves, to be set up in and moderated by the library.</p> <p>...Ooh, could amount to an extremely worthwhile internets programming project, including organizing real paper work, sort, and storage.</p> http://mathoverflow.net/questions/11591/suggestions-for-a-good-measure-theory-book/39598#39598 Answer by Martin Gisser for Suggestions for a good Measure Theory book Martin Gisser 2010-09-22T11:13:28Z 2010-09-22T11:13:28Z <p>If you want to go deeper into probability theory: I would also recommend Heinz Bauer's "Measure and integration theory". But I found it a bit dry (in German). Also Kai Lai Chung's "A Course in Probability Theory" is excellent.</p> <p>If you're more interested in (functional) analysis and want just a short intro to measue theory: As an alternative to Rudin's "Real and Complex Analysis" I warmly recommend the recent book by Jürgen Jost "Postmodern Analysis", which includes an intro to PDE. I wish I had that when I was young... Right next to these in my library is Segal &amp; Kunze "Integrals and Operators" and Robert Geroch's "Mathematical Physics" (no physics inside).</p> http://mathoverflow.net/questions/37704/are-these-operators-defined-on-2d-surfaces-self-adjoint/38408#38408 Answer by Martin Gisser for Are these operators defined on 2D surfaces self-adjoint? Martin Gisser 2010-09-11T14:03:35Z 2010-09-11T14:03:35Z <p>If H is bounded and the surface is geodesically complete, then I bet the operator is self-adjoint. The proofs of such theorems rest on an approximation of 1 by smooth compactly supported functions with bounded derivatives, constructed from the Riemannian distance function. Should be quite simple.</p> <p>I have no library access currently. Do you have a link to 2007?</p> <p>The trick should be found in one of the following (but I couldn't check).</p> <p>M.P. Gaffney, A Special Stokes's Theorem for Complete Riemannian Manifolds, Ann. of Math. (2) 60 (1954), 140 145.</p> <p>P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401 414.</p> <p>R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48 79.</p> http://mathoverflow.net/questions/54343/is-there-a-preferable-convention-for-defining-the-wedge-product/54353#54353 Comment by Martin Gisser Martin Gisser 2013-02-22T13:58:42Z 2013-02-22T13:58:42Z Why do you need the factors to be 0,1,-1? http://mathoverflow.net/questions/39540/is-cartans-magic-formula-due-to-lie-or-henri/92251#92251 Comment by Martin Gisser Martin Gisser 2012-03-28T00:50:25Z 2012-03-28T00:50:25Z @Fr&#233;d&#233;ric: Can you point out the formula in E. Goursat, Sur quelques points de la th&#233;orie des invariants int&#233;graux, J. Math. Pures Appl. (7), t. I (1915) ? A PDF copy is here: portail.mathdoc.fr/JMPA/PDF/… Andrzej Trautman (2008) writes: &quot;Around 1920, Elie Cartan defined a natural differential operator L(X) acting on fields of exterior forms. He noted that it commutes with the exterior derivative d and gave, in equation (5) on p. 84 in [2]&quot; the magic formula. Reference [2] is Lecons sur les invariants integraux. See fuw.edu.pl/~amt/4Krupka.pdf – Martin Gisser 1 hour ago http://mathoverflow.net/questions/39540/is-cartans-magic-formula-due-to-lie-or-henri/92251#92251 Comment by Martin Gisser Martin Gisser 2012-03-27T23:18:04Z 2012-03-27T23:18:04Z Alas I'm too bad at French and decyphering French math books. I'm far away from the library, but last time I was unable to identify the magic formula in E. Cartan's book. BTW the history of diff. forms seems quite messy. E.g. Poincare's lemma is due to Volterra. And Goursat called closed forms &quot;exacte&quot;. There is an article by Hans Samelson: &quot;Differential Forms, the Early Days; or the Story of Deahna's Theorem and of Volterra's Theorem&quot; Am. Math. Monthly 108 (6): 552–530 <a href="http://www.cs.sjsu.edu/faculty/beeson/courses/Ma213/HistoryOfForms.pdf" rel="nofollow">cs.sjsu.edu/faculty/beeson/courses/Ma213/&hellip;</a> No mention of the magic formula there. http://mathoverflow.net/questions/39540/is-cartans-magic-formula-due-to-lie-or-henri/92262#92262 Comment by Martin Gisser Martin Gisser 2012-03-27T22:53:04Z 2012-03-27T22:53:04Z According to an article by Andrzej Trautman (2008), the general Lie derivative on tensors got introduced by Władysław Ślebodziński (1931) and named &quot;Lie derivative&quot; (in German) by David van Dantzig (1932). Trautman article: <a href="http://www.fuw.edu.pl/~amt/4Krupka.pdf" rel="nofollow">fuw.edu.pl/~amt/4Krupka.pdf</a> http://mathoverflow.net/questions/54781/dual-riemannian-metric-and-the-dual-metric-form/54813#54813 Comment by Martin Gisser Martin Gisser 2012-02-08T13:13:46Z 2012-02-08T13:13:46Z [This is off-topic of course, but I can't resist] The (shuffle) coalgebra structure is indeed of big importance, but nobody seems to notice. Consider the &quot;total&quot; covariant derivative iterated to the k-th total covariant derivative. Apply this to a tensor product. In classical tensor calulus this gives a huge mess. But using the comultiplication you get a nice and short generalization of the general Leibniz product rule (except the binomial coefficients are now hidden in the comultiplication). Simplest nontrivial application: Curvature is a tensor. http://mathoverflow.net/questions/86666/existence-uniqueness-and-regularity-for-linear-parabolic-pde-on-a-complete-riem/86866#86866 Comment by Martin Gisser Martin Gisser 2012-02-08T01:31:48Z 2012-02-08T01:31:48Z Sorry, I'm far away from the library and won't be able to recall much more. This stuff had bugged me quite a bit - but I had a very individual approach to things (e.g. hating coordinates and Christoffel stuff and Warner's GTM94 book). I felt encouraged by a quote from Bourguignon's Weitzenb&#246;ck paper: He said, if you encounter variable coefficients, go for geometric tools. (That's contrary to your ansatz in 1st ADDED) --- What I possibly liked more was J.Roe: &quot;Elliptic Operators, topology and asymptotic methods&quot;, Pitman Res. Notes 179 (1988) --- I'm extremely curious what you will find out! http://mathoverflow.net/questions/86666/existence-uniqueness-and-regularity-for-linear-parabolic-pde-on-a-complete-riem/86866#86866 Comment by Martin Gisser Martin Gisser 2012-02-08T01:12:32Z 2012-02-08T01:12:32Z Back then I couldn't find any &quot;nice&quot; literature, so I fancied (but never worked out) something like that: 1. Use the L^2 spectral theorem. 2. Assume a compact manifold (glue small enough open subset of your general manifold in a compact one). You get a smooth map from positive reals into L^2 of course, but also into Sobolev spaces (expand powers of the Laplacian left to the exponential into powers of covariant derivative). No need for constant coefficients or a Friedrichs mollifier. But you need the Friedrichs inequality. My old notes recommend a book by E.B.Davies, Heat kernels and spct thry http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g/57713#57713 Comment by Martin Gisser Martin Gisser 2011-03-08T22:11:44Z 2011-03-08T22:11:44Z Me neither (but my math life is just a tiny fraction of yours) -- except: The K-transformed connection I mentioned in the question is a quite natural idea and it has torsion. http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g Comment by Martin Gisser Martin Gisser 2011-03-08T22:03:15Z 2011-03-08T22:03:15Z Deane, of course I don't think they arise from Killing fields. I haven't yet done the math (currently hard at work with other stuff), but methinks from the 1st Bianchi identity for curvature applied to Killing field I can just drop the antisymmetrisator and get a cyclic symmetry for the 2nd covariant derivative of the Killing field. -- Your favorite approach to Bianchi is Kazdan's &quot;another proof&quot;. My way I get them by total tensor product rules and a crucial property of the &quot;cyclator&quot;, and they look like in Cartan calculus (incl. torsion). http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g/57713#57713 Comment by Martin Gisser Martin Gisser 2011-03-08T14:29:12Z 2011-03-08T14:29:12Z Yeah. Perhaps Levi-Civita theorem would be a better name. --- I will regard my question as answered when someone can tell something about my example application (is it known, where used, ...) (but forget about coordinates - deriving a coordinate formula that way is of course not the least painful way...) http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g Comment by Martin Gisser Martin Gisser 2011-03-08T14:20:31Z 2011-03-08T14:20:31Z My latest speculatory argument turned out to be complete nonsense... But still methinks I smell something with Killing fields. The Petersen Koszul formula says the covariant derivative of a Killing 1-form-from-field is the exterior differential. (So the point seems more: Koszul's formula has nothing to say on Killing fields because it gives essentially zero.) Thru the lens of my calculus I see Bianchi identity symmetries lurking behind covariant differentiation of Killing fields. http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g/57713#57713 Comment by Martin Gisser Martin Gisser 2011-03-08T11:44:57Z 2011-03-08T11:44:57Z Haha, good question. My few books call it the fundamental theorem: Existence and uniqueness of the Levi-Civita connection. -- I have currently no idea why I called it lemma. http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g/57713#57713 Comment by Martin Gisser Martin Gisser 2011-03-08T03:24:54Z 2011-03-08T03:24:54Z P.S.: Said German differential ring theorist's lectures had at least a complete proof of the fundamental lemma of Riem. Geom., completely proving tensoriality on the r.h.s. P.P.S.: My answer-comment to Jose is hidden (anyhow syntacticly mangled) - Killing fields indeed serving an example of Koszul vs. not-Koszul. P.P.P.S.: Something learned today. Thanks, sirs. http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g/57713#57713 Comment by Martin Gisser Martin Gisser 2011-03-08T03:05:33Z 2011-03-08T03:05:33Z Deane, 1) my example even involves 2 connections... 2a) I find it disadvantageous to have to learn at least 3 calculi for the whole picture. 2b) I found me developing my own calculus to sanely get into Hilbert space via Stokes theorem (for doing geometric PDE geometrically, like Bochner theorems) Indeed Jos&#233; has an excellent answer. There's sure much more. Alas I had to suffer through O'Neill's formulae etc. in Christoffelian (by a famous German differential ring theorist, no less). http://mathoverflow.net/questions/57649/applications-of-koszuls-formula-other-than-the-fundamental-lemma-of-riemannian-g Comment by Martin Gisser Martin Gisser 2011-03-08T02:56:47Z 2011-03-08T02:56:47Z Jos&#233;, I guess you mean by Killing's equation Pepeter Petersen, Riem. Geom. 2nd ed, Proposition 27? Yeah, I should have had a 2nd look in this book, having also O'Neill and Mixed Curvature Equations with Koszul (KF). - Excuse given to Deane below --- I almost bet Petersen's Proposition 28 could also be proved with KF, showing that $f\mapsto\nabla_V(fX)$ is tensorial if $X\vert_p=0 and$\nablaX\vert_p\$. So his proof would be a &quot;bad&quot; example for my funny conjecture that &lt;i&gt;anything you can derive from a KF&lt;/i&gt; can also be derived without Lie brackets. Yeah the paradox - but now I go sleep...