Twisted Hessian curves: Difference between revisions

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In mathematics, a '''recurrent tensor''', with respect to a [[connection (mathematics)|connection]] <math>\nabla</math> on a [[manifold]] ''M'', is a [[Tensor field|tensor]] ''T'' for which there is a [[differential form|one-form]] ''ω'' on ''M'' such that
 
:<math>\nabla T = \omega\otimes T. \, </math>
 
==Examples==
 
===Parallel Tensors===
An example for recurrent [[Tensor field|tensor]]s are parallel tensors which are defined by
:<math>\nabla A = 0 </math>
with respect to some connection <math>\nabla</math>.
 
If we take a [[pseudo-Riemannian manifold]] <math>(M,g)</math> then the metric ''g''  is a parallel and therefore recurrent tensor with respect to its [[Levi-Civita connection]], which is defined via
:<math>\nabla^{LC} g = 0 </math>
and its property to be torsion-free.
 
Parallel vector fields (<math>\nabla X = 0</math>) are examples of recurrent tensors that find importance in mathematical research. For example, if <math> X </math> is a recurrent non-null vector field on a [[pseudo-Riemannian manifold]] satisfying
:<math>\nabla X = \omega\otimes X </math>
for some closed [[one-form]] <math> \omega </math>, then X can be rescaled to a parallel vector field.<ref>Alekseevsky, Baum (2008)</ref> In particular, non-parallel recurrent vector fields are null vector fields.
 
===Metric space===
Another example appears in connection with [[Weyl structure]]s. Historically, Weyl structures emerged from the considerations of [[Hermann Weyl]] with regards to properties of parallel transport of vectors and their length.<ref>Weyl (1918)</ref> By demanding that a manifold have an affine parallel transport in such a way that the manifold is locally an [[affine space]], it was shown that the induced connection had a vanishing torsion tensor
:<math>T^\nabla(X,Y) = \nabla_XY-\nabla_YX - [X,Y] = 0</math>.
Additionally, he claimed that the manifold must have a particular parallel transport in which the ratio of two transported vectors is fixed. The corresponding connection <math>\nabla'</math> which induces such a parallel transport satisfies
:<math>\nabla' g = \varphi \otimes g</math>
for some one-form <math>\varphi</math>. Such a metric is a recurrent tensor with respect to <math>\nabla'</math>. As a result, Weyl called the resulting manifold <math>(M,g)</math> with affine connection <math>\nabla</math> and recurrent metric <math> g </math> a metric space. In this sense, Weyl was not just referring to one metric but to the conformal structure defined by <math> g </math>.  
 
Under the conformal transformation <math>g \rightarrow e^{\lambda}g</math>, the form <math>\phi</math> transforms as <math>\varphi \rightarrow \varphi -d\lambda</math>. This induces a canonical map <math>F:[g] \rightarrow \Lambda^1(M)</math> on <math>(M, [g])</math> defined by
:<math>F(e^\lambda g) := \varphi - d\lambda</math>,
where <math>[g]</math> is the conformal structure. <math>F</math> is called a Weyl structure,<ref>Folland (1970)</ref> which more generally is defined as a map with property
:<math>F(e^\lambda g) = F(g) - d\lambda</math>.
 
===Recurrent spacetime===
One more example of a recurrent tensor is the curvature tensor <math>\mathcal{R}</math> on a recurrent spacetime,<ref>Walker (1948)</ref> for which
:<math>\nabla \mathcal{R} = \omega \otimes \mathcal{R}</math>.
 
==References==
<references/>
 
==Literature==
*{{cite journal |author=Weyl, H. |title=Gravitation und Elektrizität |journal=Sitzungsberichte der preuss. Akad. d. Wiss. |year=1918 |pages=465}}
* [http://qjmath.oxfordjournals.org/cgi/reprint/os-20/1/135.pdf A.G. Walker: ''On parallel fields of partially null vector spaces''], The Quarterly Journal of Mathematics 1949, Oxford Univ. Press
* [http://qjmath.oxfordjournals.org/cgi/reprint/2/1/151.pdf E.M. Patterson: ''On symmetric recurrent tensors of the second order''], The Quarterly Journal of Mathematics 1950, Oxford Univ. Press
*[http://www.jstor.org/stable/1993404 J.-C. Wong: ''Recurrent Tensors on a Linearly Connected Differentiable Manifold''], Transactions of the American Mathematical Society 1961,
* [http://www.intlpress.com/JDG/archive/1970/4-1&2-145.pdf G.B. Folland: ''Weyl Manifolds''], J. Differential Geometry 1970
*{{cite book | author=D.V. Alekseevky, H. Baum| title = Recent developments in pseudo-Riemannian geometry | publisher=European Mathematical Society | year=2008 |isbn = 3-03719-051-5}}
 
{{DEFAULTSORT:Recurrent Tensor}}
[[Category:Riemannian geometry]]
[[Category:Tensors]]

Latest revision as of 13:39, 5 May 2014

These fuzzies may be a small annoyance, but they can make your gym seem unkempt. Normal pedaling motion disengages the brake cane, allowing for normal pedaling. This temporarily blocks penile blood flow causing impotence. Riding position is typically more vertical than that of a road bike. In case you loved this post and you would love to receive much more information relating to Women mountain bike sizing. assure visit our own web-page. Certain Haro mountain bike models worth mentioning right here consist of BMX Black trail X1, BMX Haro Partial 16, BMX Haro FIC and so forth.

It'll have you on the edge of your seat and holding your breath for Zink to land this record-breaking backflip. "Before breathing exercises it took me 45 minutes to really get into my stride when off road mountain biking. If you don't do a lot of racing on mountain trails then you need a road bike. Just to give you an idea, mine has three settings, those being; *Completely off, so that I am riding it as a normal bike; *Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). Browse a number of the on-line forums to determine what other riders like and do not like about their bicycles.

Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop. An Indian travois style trailer is very simple to make and very effective for heavy loads. Please remember the stem lengths can impact just how reactive the bike is. Just like a rectangle you have 2 long sides and 2 short sides. The most common sized crankset is the 170 millimeter.

Consequently, they became like villains in the eyes of the NBA fans outside Miami. Mountain Bike shoes are available in many varieties and options. This article is to help you when buying a new bike. There is no shortage of videos of mountain bike crashes from around the world, with many riders with appropriate safety equipment escaping serious injury, while others were not that fortunate. Within 2 weeks of breathing exercises, the warm up period disappeared.

(many commute bikes come with front suspension and disc brakes these days). Then, the answer is the skeletal frame size that will fit you for a normal bike. And the cyclist may certainly sense the distinction. Sign up to get a mountain bike expedition and relive people childhood biking wonder and thrills with a whole new tier. The popular Whistler mountain bike website now has an extension into iphone.

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