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In [[Riemannian geometry]], the '''sectional curvature''' is one of the ways to describe the [[curvature of Riemannian manifolds]]. The sectional curvature ''K''(&sigma;<sub>''p''</sub>) depends on a two-dimensional plane &sigma;<sub>''p''</sub> in the tangent space at ''p''. It is the [[Gaussian curvature]] of the [[surface]] which has the plane &sigma;<sub>''p''</sub> as a tangent plane at ''p'', obtained from [[geodesic]]s which start at ''p'' in the directions of &sigma;<sub>''p''</sub> (in other words, the image of &sigma;<sub>''p''</sub> under the [[exponential map]] at ''p''). The sectional curvature is a smooth real-valued function on the 2-[[Grassmannian]] [[fiber bundle|bundle]] over the manifold.
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The sectional curvature determines the [[Riemann curvature tensor|curvature tensor]] completely.
 
==Definition==
Given a [[Riemannian manifold]] and two [[linearly independent]] [[tangent vectors]] at the same point, ''u'' and ''v'', we can define
 
:<math>K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2}</math>
 
Here ''R'' is the [[Riemann curvature tensor]].
 
In particular, if ''u'' and ''v'' are [[orthonormal]], then
:<math>K(u,v) = \langle R(u,v)v,u\rangle.</math>
The sectional curvature in fact depends only on the 2-plane σ<sub>''p''</sub> in the tangent space at ''p'' spanned by ''u'' and ''v''. It is called the '''sectional curvature of the 2-plane &sigma;<sub>''p''</sub>''', and is denoted ''K''(&sigma;<sub>''p''</sub>).
 
==Manifolds with constant sectional curvature==
 
[[Riemannian manifold]]s with constant sectional curvature are the most simple. These are called [[space form]]s.  By rescaling the metric there are three possible cases
*negative curvature &minus;1, [[hyperbolic geometry]]
*zero curvature, [[Euclidean geometry]]
*positive curvature +1, [[elliptic geometry]]
The model manifolds for the three geometries are [[hyperbolic space]], [[Euclidean space]] and a unit [[n-sphere|sphere]]. They are the only [[Complete space|complete]], [[simply connected]] Riemannian manifolds of given sectional curvature.  All other connected complete constant curvature manifolds are quotients of those by some group of [[isometry|isometries]].
 
If for each point in a connected Riemannian manifold (of dimension three or greater) the sectional curvature is independent of the tangent 2-plane, then the sectional curvature is in fact constant on the whole manifold.
 
==Toponogov's theorem==
[[Toponogov's theorem]] affords a characterization of sectional curvature in terms of how "fat" geodesic triangles appear when compared to their Euclidean counterparts.  The basic intuition is that, if a space is positively curved, then the edge of a triangle opposite some given vertex will tend to bend away from that vertex, whereas if a space is negatively curved, then the opposite edge of the triangle will tend to bend towards the vertex.
 
More precisely, let ''M'' be a [[complete space|complete]] Riemannian manifold, and let ''xyz'' be a geodesic triangle in ''M'' (a triangle each of whose sides is a length-minimizing geodesic). Finally, let ''m'' be the midpoint of the geodesic ''xy''. If ''M'' has non-negative curvature, then for all sufficiently small triangles
:<math>d(z,m)^2\ge \tfrac12d(z,x)^2 + \tfrac12d(z,y)^2 - \tfrac14d(x,y)^2</math>
where ''d'' is the [[distance function]] on ''M''. The case of equality holds precisely when the curvature of ''M'' vanishes, and the right-hand side represents the distance from a vertex to the opposite side of a geodesic triangle in Euclidean space having the same side-lengths as the triangle ''xyz''. This makes precise the sense in which triangles are "fatter" in positively curved spaces. In non-positively curved spaces, the inequality goes the other way:
:<math>d(z,m)^2\le \tfrac12d(z,x)^2 + \tfrac12d(z,y)^2 - \tfrac14d(x,y)^2.</math>
 
If tighter bounds on the sectional curvature are known, then this property generalizes to give a [[comparison theorem]] between geodesic triangles in ''M'' and those in a suitable simply connected space form; see [[Toponogov's theorem]]. Simple consequences of the version stated here are:
 
*A complete Riemannian manifold has non-negative sectional curvature if and only if the function <math>f_p(x)=\operatorname{dist}^2(p,x)</math> is 1-[[Glossary of Riemannian and metric geometry|concave]] for all points ''p''.
*A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function <math>f_p(x)=\operatorname{dist}^2(p,x)</math> is 1-[[Glossary of Riemannian and metric geometry|convex]].
 
==Manifolds with non-positive sectional curvature==
In 1928, [[Élie Cartan]] proved the [[Cartan–Hadamard theorem]]: if ''M'' is a [[Complete space|complete]] manifold with non-positive sectional curvature, then its [[universal cover]] is [[diffeomorphic]] to a [[Euclidean space]]. In particular, it is [[Aspherical space|aspherical]]: the [[homotopy groups]] <math>\pi_i(M)</math> for ''i'' &ge; 2 are trivial.  Therefore, the topological structure of a complete non-positively curved manifold is determined by its [[fundamental group]].  [[Preissman's theorem]] restricts the fundamental group of negatively curved compact manifolds.
 
==Manifolds with positive sectional curvature==
Little is known about the structure of positively curved manifolds. The [[soul theorem]] ({{harvnb|Cheeger|Gromoll|1972}}; {{harvnb|Gromoll|Meyer|1969}}) implies that a complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold. As for compact positively curved manifolds, there are two classical results:
 
*It follows from the [[Myers theorem]] that the fundamental group of such manifold is finite.
 
*It follows from the [[Synge theorem]] that the fundamental group of such manifold in even dimensions is 0, if orientable and <math>\Bbb Z_2</math> otherwise. In odd dimensions a positively curved manifold is always orientable.
 
Moreover, there are relatively few examples of compact positively curved manifolds, leaving a lot of conjectures (e.g., the [[Hopf conjecture]] on whether there is a metric of positive sectional curvature on <math>\Bbb S^2\times\Bbb S^2</math>). The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if <math>(M,g)</math> is a Riemannian manifold admitting a free isometric action of a Lie group G, and M has positive sectional curvature on all 2-planes orthogonal to the orbits of G, then the manifold <math>M/G</math> with the quotient metric has positive sectional curvature. This fact allows one to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples {{harv|Ziller|2007}}:
 
*The Berger spaces <math>B^7=SO(5)/SO(3)</math> and <math>B^{13}=SU(5)/Sp(2)\cdot\Bbb S^1</math>.
 
*The Wallach spaces (or the homogeneous flag manifolds): <math>W^6=SU(3)/T^2</math>, <math>W^{12}=Sp(3)/Sp(1)^3</math> and <math>W^{24}=F_4/Spin(8)</math>.
 
*The Aloff–Wallach spaces <math>W^7_{p,q}=SU(3)/\operatorname{diag}(z^p,z^q,\overline{z}^{p+q})</math>.
 
*The Eschenburg spaces <math>E_{k,l}=\operatorname{diag}(z^{k_1},z^{k_2},z^{k_3})\backslash SU(3)/\operatorname{diag}(z^{l_1},z^{l_2},z^{l_3})^{-1}.</math>
 
*The Bazaikin spaces <math>B^{13}_p=\operatorname{diag}(z_1^{p_1},\dots,z_1^{p_5})\backslash U(5)/\operatorname{diag}(z_2A,1)^{-1}</math>, where <math>A\in Sp(2)\subset SU(4)</math>.
 
==References==
*{{Citation | doi=10.2307/1970819 | last1=Cheeger | first1=Jeff | last2=Gromoll | first2=Detlef | title=On the structure of complete manifolds of nonnegative curvature | mr=0309010 | year=1972 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]]  | volume=96 | pages=413–443 | issue=3 | publisher=Annals of Mathematics | jstor=1970819}}.
*{{Citation | doi=10.2307/1970682 | last1=Gromoll | first1=Detlef | last2=Meyer | first2=Wolfgang | title=On complete open manifolds of positive curvature | mr=0247590 | year=1969 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]]  | volume=90 | pages=75–90 | issue=1 | publisher=Annals of Mathematics | jstor=1970682}}.
* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Morse theory | publisher=[[Princeton University Press]] | series=Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 | mr=0163331 | year=1963}}.
* {{Citation | last1=Petersen | first1=Peter | title=Riemannian geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-29246-5 | mr=2243772 | year=2006 | volume=171}}.
*{{cite arxiv|first=Wolfgang|last=Ziller|title=Examples of manifolds with non-negative sectional curvature|eprint=math/0701389|year=2007}}.
 
==See also==
 
*[[Riemann curvature tensor]]
*[[curvature of Riemannian manifolds]]
*[[curvature]]
 
{{curvature}}
 
[[Category:Riemannian geometry]]
[[Category:Curvature (mathematics)]]

Latest revision as of 17:52, 4 January 2015

Okay so exactly where do you go to get the best massage? Of course you can seek out an appointment for your self someplace, or you can discover a very good location for your partner and you to go collectively. Nonetheless, would not you like to be able to give the very best massage for your partner? Maintain reading to find out how to do that.

When you are giving a massage, make confident that you are obtaining your thumbs into the mix. Strong and strong, your thumbs can actually dig in. Press gently, as you do not want to make folks feel uncomfortable.

If you endure from frequent tension headaches, you might benefit from a expert deep tissue neck massage. To perform a deep tissue massage, the massage therapist utilizes a stretching method along with stress to pull and stretch your muscle tissues. This permits the muscle tissues to relax thus, relieving your headache and tension.

Are you prone to muscle cramps either brought on by physical activity or occasionally absolutely nothing at all? These can be extremely painful, specially when they are in your calf. Massages can assist to prevent your muscle tissues from cramping up by advertising blood flow. You can also massage the area as quickly as you really feel it starting to cramp up for some quick relief.

The raking technique is a well-liked way to give a back massage. Spread your fingers out wide and use your fingertips to perform a raking motion down the back, from shoulders to hips. Be taught further about almira masaj salonu antalya by browsing our striking article. You can increase the pressure to the recipient"s liking. This is a easy way to unwind your significant other and devote some good quality time with each other.

Make sure you ask concerns when acquiring a massage. Dumb inquiries do not exist and the masseuse ought to be thrilled to give you answers. Your relaxation is the finish aim, so ask away.

If you have a wound that is beginning to heal, massaging around that area will support it to heal even faster. Massage increases the amount of blood that is flowing to that area of your physique. So, by massaging that location you are providing it with new blood to market healing. This can also assist to decrease scars.

Asian massages are identified to support loosen up the physique. One not-so-common, but really relaxing Asian massage is named the Indonesian-Javanese massage. With this approach, the therapist will use both their hands and knuckles to get so deep into the skin. This will support your muscles loosen up if they have been tense.

Support the massage therapist by relaxing your physique. Avoid trying to "aid" by tensing specific muscles or regions. The massage therapist is nicely-educated in various methods to maneuver your physique components to achieve the best results from massage, but their perform is hindered if you cannot loosen up. Try performing some deep breathing and practice just letting go and trusting the therapist to do a fantastic job.

Anytime you get oneself a massage, never be afraid to inform your massage therapist of your issue regions. A massage"s aim is to relax your muscle tissues and target your dilemma places. That is why it"s a excellent notion to be very clear with your therapist about any aches, pains or other difficulties you have been having.

Be vocal with your massage therapist.Speaking could be the final factor you really want to do for the duration of a massage, but it can be the distinction among an ok experience and a fantastic one particular. You need to have to let the therapist know what locations want perform and what methods perform greatest.

There is no true dress code when you decide to go out and get a massage. If you are concerned about dressing proper, speak with the spa or your message therapist beforehand. By addressing clothing issues, you can guarantee that you won"t feel embarrassed or uncomfortable for the duration of your massage. Get further on this affiliated essay - Click here: antalya kardelen masaj salonu ariyorsaniz sitemize tiklayin - YouTube. If there is an region you also don"t want massaged like your behind, inform the therapist just before beginning.

Communicate with your massage therapist. For a second viewpoint, consider glancing at: understandable. Your masseuse is most most likely not a thoughts reader, so let them know if an area of your physique wants a tiny added interest. Regardless of whether your shoulders are a mess from becoming hunched over a desk or your legs are sore from a marathon exercise, speak up.

How lengthy must you massage a spot for? Try to aim for about thirty seconds. If you have many locations to rub, this will assist to speed along the approach. If you locate that the spot nevertheless aches or is tight, return to it later for another 30 second session.

As an alternative of sending yourself or your companion off to a massage clinic, now you know how to give the ideal massage your self. Of course you can not do your personal, but you can show your partner that he or she can just as easily discover how to give one particular as effectively. Get began and show them!.

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