Introduction In most presentations of Riemannian geometry, e.g. O’Neill (1983) and Wikipedia, the fundamental theorem of Riemannian geometry (“the miracle of Riemannian geometry”) is given: that for any semi-Riemannian manifold there is a unique torsion-free metric connection. I assume partly because of this and partly because the major application of Riemannian geometry is General Relativity, … Continue reading Mercator: A…
#semi-riemannian manifolds
11 posts
24 Nov 2016
17 Apr 2016
Introduction In their paper Betancourt et al. (2014), the authors give a corollary which starts with the phrase “Because the manifold is paracompact”. It wasn’t immediately clear why the manifold was paracompact or indeed what paracompactness meant although it was clearly something like compactness which means that every cover has a finite sub-cover. It turns … Continue reading Every Manifold…
14 Feb 2016
Introduction In proposition 58 Chapter 1 in the excellent book O’Neill (1983), the author demonstrates that the Lie derivative of one vector field with respect to another is the same as the Lie bracket (of the two vector fields) although he calls the Lie bracket just bracket and does not define the Lie derivative preferring … Continue reading The Lie…
26 Jul 2009
O’Neill 2.6: from 2.13, we know that . Suppose that . Thus . And so we must have as required.
29 Oct 2008
On page 19, O’Neill comments that the proof of Lemma 33 is a mild generalization of the proof of proposition 28. I think (2) (3) requires spelling out. Let and let be a co-ordinate system at . Let be a co-ordinate system at . Then by (2) has rank . Thus by exercise 7 and … Continue reading Immersions
18 May 2008
Let be a curve and be an element of the set of all smooth vector fields on . Let re-parameterize the curve. Then and . Thus which is smooth since the composition of smooth functions is a smooth function and therefore . By definition So Also by definition So, using the chain rule Which gives … Continue reading (Induced) Covariant…
5 May 2008
2 May 2008
Define . This is well defined since if is another bump function then and reversing and gives
26 Apr 2008
Let and be sub-manifolds of . Then and there are charts: for about adapted to for about adpapted to such that Now let be the injection map. Consider the commutative diagram below. Then which is smooth. Hence is smooth.
22 Apr 2008
Let be a vector space with linear isomorphisms to as charts. Let be a curve . Then and . But is linear (see the exercise) and so as was required to be shown.
17 Apr 2008
Let be integral curves in a manifold then is closed. Proof: Let be have a limit in . Let be a chart then converges to by continuity. But also converges to . Hence and so . Thus is closed.