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In [[mathematics]], a '''sub-Riemannian manifold''' is a certain type of generalization of a [[Riemannian manifold]]. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal subspaces''.
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Sub-Riemannian manifolds (and so, ''a fortiori'', Riemannian manifolds) carry a natural [[intrinsic metric]] called the '''metric of Carnot–Carathéodory'''. The [[Hausdorff dimension]] of such [[metric space]]s is always an [[integer]] and larger than its [[topological dimension]] (unless it is actually a Riemannian manifold).
 
Sub-Riemannian manifolds often occur in the study of constrained systems in [[classical mechanics]], such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the [[Berry phase]] may be understood in the language of sub-Riemannian geometry. The [[Heisenberg group]], important to [[quantum mechanics]], carries a natural sub-Riemannian structure.
 
==Definitions==
 
By a ''distribution'' on <math>M</math> we mean a [[subbundle]] of the [[tangent bundle]] of  <math>M</math>.
Given a distribution <math>H(M)\subset T(M)</math> a vector field in <math>H(M)\subset T(M)</math> is called '''horizontal'''. A curve <math>\gamma</math> on <math>M</math> is called '''horizontal''' if  <math>\dot\gamma(t)\in H_{\gamma(t)}(M)</math> for any  
<math>t</math>.
 
A distribution on <math>H(M)</math> is called '''completely non-integrable''' if for any <math>x\in M</math> we have that any tangent vector can be presented as a [[linear combination]] of vectors of the following types <math>A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc\in T_x(M)</math> where all vector fields <math>A,B,C,D, \dots</math> are horizontal.
 
A '''sub-Riemannian manifold''' is a triple <math>(M, H, g)</math>, where <math>M</math> is a differentiable [[manifold]], <math>H</math> is a ''completely non-integrable'' "horizontal" distribution and <math>g</math> is a smooth section of positive-definite [[quadratic form]]s on <math>H</math>.
 
Any  '''sub-Riemannian manifold''' carries the natural [[intrinsic metric]], called the '''metric of Carnot–Carathéodory''', defined as
:<math>d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))} \, dt,</math>
where infimum is taken along all ''horizontal curves'' <math>\gamma: [0, 1] \to M</math> such that <math>\gamma(0)=x</math>, <math>\gamma(1)=y</math>.
 
==Examples==
 
A position of a car on the plane is determined by three parameters: two coordinates <math>x</math> and <math>y</math> for the location and an angle <math>\alpha</math> which describes the orientation of the car. Therefore, the position of car can be described by a point in a manifold
:<math>\mathbb R^2\times S^1.</math>
 
One can ask, what is the minimal distance one should drive to get from one position to another? This defines a [[Carnot–Carathéodory metric]] on the manifold
:<math>\mathbb R^2\times S^1.</math>
 
A closely related example of a sub-Riemannian metric can be constructed on a [[Heisenberg group]]: Take two elements <math>\alpha</math> and <math>\beta</math> in the corresponding Lie algebra such that  
:<math>\{ \alpha,\beta,[\alpha,\beta]\}</math>
 
spans the entire algebra. The horizontal distribution <math>H</math> spanned by left shifts of <math>\alpha</math> and <math>\beta</math> is ''completely non-integrable''. Then choosing any smooth positive quadratic form on <math>H</math> gives a sub-Riemannian metric on the group.
 
==Properties==
 
For every sub-Riemannian manifold, there exists a [[Hamiltonian mechanics|Hamiltonian]], called the '''sub-Riemannian Hamiltonian''', constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold. The existence of geodesics of the corresponding [[Hamilton–Jacobi equation]]s for the sub-Riemannian Hamiltonian is given by the [[Chow–Rashevskii theorem]].
 
==See also==
*[[Carnot group]], a class of [[Lie group]]s that form sub-Riemannian manifolds
 
==References==
 
*{{Citation | editor1-last=Bellaïche | editor1-first=André | editor2-last=Risler | editor2-first=Jean-Jacques | title=Sub-Riemannian geometry | url=http://books.google.com/books?id=7Z7IMze7pDwC | publisher=Birkhäuser Verlag | series=Progress in Mathematics | isbn=978-3-7643-5476-3 | mr=1421821 | year=1996 | volume=144}}
*{{Citation | last1=Gromov | first1=Mikhael | editor1-last=Bellaïche | editor1-first=André | editor2-last=Risler. | editor2-first=Jean-Jacques | title=Sub-Riemannian geometry | url=http://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progr. Math. | mr=1421823 | year=1996 | volume=144 | chapter=Carnot-Carathéodory spaces seen from within | pages=79–323 | isbn=3-7643-5476-3}}
*{{citation|url=http://www.math.ethz.ch/~ledonnee/sub-Riem_notes.pdf |title=Lecture notes on sub-Riemannian geometry|first=Enrico|last= Le Donne}}
* Richard Montgomery, ''A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91)'', (2002) American Mathematical Society, ISBN 0-8218-1391-9.
 
[[Category:Metric geometry]]
[[Category:Riemannian manifolds]]

Revision as of 04:10, 5 March 2014

If your computer is running slow, you have probably gone from the different stages of rage and frustration. Having such a awesome tool like a computer could seem like a curse plus a blessing at the same time whenever this occurs. It is amazing whenever it is actually running swiftly plus smooth, however, then once it starts acting weird and slows way down, frustration sets in. How may anything because fantastic because a computer make a person thus mad?

We all learn that the registry is the important component of the Windows operating system because it shops all information about the Dll files, programs found on the computer and system settings. However, because days by, it is very unavoidable which you might encounter registry matter due to a big amount of invalid, useless plus unwanted entries.

The Windows registry is a system database of info. Windows and different software shop a lot of settings and additional information in it, plus retrieve such info from the registry all time. The registry is furthermore a bottleneck in which considering it's the heart of the running program, any difficulties with it can result mistakes plus bring the running program down.

The issue with many of the persons is the fact that they do not wish To invest income. In the cracked adaptation 1 does not have to pay anything and can download it from web especially easily. It is simple to install also. However, the issue comes whenever it is not able to identify all possible viruses, spyware and malware in the system. This is because it happens to be obsolete in nature and does not get any usual changes within the webpage downloaded. Thus, a system is accessible to problems like hacking.

The final step is to make sure which you clean the registry of your computer. The "registry" is a big database that shops important files, settings & options, plus information. Windows reads the files it requires inside purchase for it to run programs through this database. If the registry gets damaged, infected, or clogged up, then Windows will likely not be able to correctly access the files it needs for it to load up programs. As this arises, difficulties and errors like the d3d9.dll error occur. To fix this and avoid future setbacks, you must download and run a registry cleaning tool. The highly recommended software is the "Frontline tuneup utilities 2014".

Software mistakes or hardware mistakes which happen whenever running Windows plus intermittent mistakes are the general factors for a blue screen bodily memory dump. New software or motorists that have been installed or changes inside the registry settings are the typical s/w causes. Intermittent mistakes refer to failed program memory/ difficult disk or over heated processor and these too could result the blue screen bodily memory dump error.

As the hub center of the computer, all of the significant settings are stored the registry. Registry is structured as keys plus every key relates to a program. The program reads the keys plus utilizes the information to launch and run programs. However, the big problem is that there are too countless unwanted settings, useless info occuping the useful area. It makes the program run gradually plus big amounts of settings become unreadable.

Fortunately, there's a simple technique to fix most the computer errors. You really should be able to fix corrupt registry files on the computer. And to do that, we could just use a tool termed as a registry cleaner. These simple pieces of software actually scan from the PC plus fix each corrupt file that may result a issue to Windows. This enables your computer to employ all of the files it wants, which not just speeds it up - nevertheless also stops all mistakes on your program too.

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