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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''(σ<sub>''p''</sub>) depends on a two-dimensional plane σ<sub>''p''</sub> in the tangent space at ''p''. It is the [[Gaussian curvature]] of the [[surface]] which has the plane σ<sub>''p''</sub> as a tangent plane at ''p'', obtained from [[geodesic]]s which start at ''p'' in the directions of σ<sub>''p''</sub> (in other words, the image of σ<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. | |||
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 σ<sub>''p''</sub>''', and is denoted ''K''(σ<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 −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'' ≥ 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)]] | |||
Revision as of 04:05, 13 January 2014
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional plane σp in the tangent space at p. It is the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp (in other words, the image of σp under the exponential map at p). The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.
The sectional curvature determines the curvature tensor completely.
Definition
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define
Here R is the Riemann curvature tensor.
In particular, if u and v are orthonormal, then
The sectional curvature in fact depends only on the 2-plane σp in the tangent space at p spanned by u and v. It is called the sectional curvature of the 2-plane σp, and is denoted K(σp).
Manifolds with constant sectional curvature
Riemannian manifolds with constant sectional curvature are the most simple. These are called space forms. By rescaling the metric there are three possible cases
- negative curvature −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 sphere. They are the only complete, simply connected Riemannian manifolds of given sectional curvature. All other connected complete constant curvature manifolds are quotients of those by some group of 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 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
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:
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 is 1-concave for all points p.
- A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function is 1-convex.
Manifolds with non-positive sectional curvature
In 1928, Élie Cartan proved the Cartan–Hadamard theorem: if M is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space. In particular, it is aspherical: the homotopy groups for i ≥ 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 (Template:Harvnb; Template:Harvnb) 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 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 ). The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if 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 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 Template:Harv:
- The Berger spaces and .
- The Wallach spaces (or the homogeneous flag manifolds): , and .
- The Aloff–Wallach spaces .
- The Eschenburg spaces
- The Bazaikin spaces , where .
References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Template:Cite arxiv.