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In mathematics, Reeb sphere theorem, named after Georges Reeb, states that

A closed oriented connected manifold Mⁿ that admits a singular foliation having only centers is homeomorphic to the sphere Sⁿ and the foliation has exactly two singularities.

Morse foliation

A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are levels of a Morse function, being the singularity a critical point of the function. The singularity is a center if it is a local extremum of the function; otherwise, the singularity is a saddle.

The number of centers c and the number of saddles $s$ , specifically c − s, is tightly connected with the manifold topology.

We denote ind p = min(k, n − k), the index of a singularity $p$ , where k is the index of the corresponding critical point of a Morse function. In particular, a center has index 0, index of a saddle is at least 1.

A Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class C² with isolated singularities such that:

each singularity of F is of Morse type,
each singular leaf L contains a unique singularity p; in addition, if ind p = 1 then $L ∖ p$ is not connected.

Reeb sphere theorem

This is the case c > s = 0, the case without saddles.

Theorem:^[1] Let $M^{n}$ be a closed oriented connected manifold of dimension $n \geq 2$ . Assume that $M^{n}$ admits a $C^{1}$ -transversely oriented codimension one foliation $F$ with a non empty set of singularities all of them centers. Then the singular set of $F$ consists of two points and $M^{n}$ is homeomorphic to the sphere $S^{n}$ .

It is a consequence of the Reeb stability theorem.

Generalization

More general case is $c > s \geq 0 .$

In 1978, E. Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles. He showed that the number of centers cannot be too much as compared with the number of saddles, notably, $c \leq s + 2$ . So there are exactly two cases when $c > s$ :

(1)

c = s + 2,

(2)

c = s + 1 .

He obtained a description of the manifold admitting a foliation with singularities that satisfy (1).

Theorem:^[2] Let $M^{n}$ be a compact connected manifold admitting a Morse foliation $F$ with $c$ centers and $s$ saddles. Then $c \leq s + 2$ . In case $c = s + 2$ ,

$M$ is homeomorphic to $S^{n}$ ,

all saddles have index 1,

each regular leaf is diffeomorphic to $S^{n - 1}$ .

Finally, in 2008, C. Camacho and B. Scardua considered the case (2), $c = s + 1$ . Interestingly, this is possible in a small number of low dimensions.

Theorem:^[3] Let $M^{n}$ be a compact connected manifold and $F$ a Morse foliation on $M$ . If $s = c + 1$ , then

$n = 2, 4, 8$ or $16$ ,

$M^{n}$ is an Eells–Kuiper manifold.

References

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↑ 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.

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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.

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[1] 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.

[2] 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.

[3] 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.

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+In [[mathematics]], '''Reeb sphere theorem''', named after [[Georges Reeb]], states that
+: A closed oriented connected manifold ''M''<sup>&nbsp;''n''</sup> that admits a [[singular foliation]] having only centers  is [[homeomorphism|homeomorphic]] to the [[sphere]] ''S''<sup>''n''</sup> and the foliation has exactly two singularities.
+==Morse foliation==
+A singularity of a foliation ''F'' is of '''Morse type''' if in its small neighborhood all leaves of the foliation are  [[level set|levels]] of a [[Morse theory|Morse function]], being the singularity a [[Critical point (mathematics)|critical point]] of the function.  The singularity is a '''center''' if it is a [[maxima and minima|local extremum]] of the function; otherwise, the singularity is a [[saddle]].
+The number of centers ''c'' and the number of saddles <math>s</math>, specifically ''c''&nbsp;&minus;&nbsp;''s'', is tightly connected with the manifold topology.
+We denote ind&nbsp;''p''&nbsp;=&nbsp;min(''k'',&nbsp;''n''&nbsp;&minus;&nbsp;''k''), the '''index''' of a singularity <math>p</math>, where ''k'' is the index of the corresponding critical point of a Morse function. In particular, a center has index&nbsp;0, index of a saddle is at least&nbsp;1.
+A '''Morse foliation''' ''F'' on a manifold ''M'' is a [[singular foliation|singular]] transversely oriented codimension one foliation of class ''C''<sup>2</sup> with isolated singularities such that:
+* each singularity of ''F'' is of Morse type,
+* each [[singular leaf]] ''L'' contains a unique singularity&nbsp;''p''; in addition, if ind&nbsp;''p''&nbsp;=&nbsp;1 then <math>L\setminus p</math> is not connected.
+==Reeb sphere theorem==
+This is the case ''c''&nbsp;>&nbsp;''s''&nbsp;=&nbsp;0, the case without saddles.
+Theorem:<ref>{{citation
+ | last = Reeb | first = Georges | author-link = Georges Reeb
+ | journal = [[Comptes rendus de l'Académie des sciences|C. R. Acad. Sci. Paris]]
+ | language = French
+ | mr = 0015613
+ | pages = 847–849
+ | title = Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique
+ | volume = 222
+ | year = 1946}}.</ref> ''Let <math>M^n</math> be a closed oriented connected manifold of dimension <math>n\ge 2</math>. Assume that <math>M^n</math> admits a <math>C^1</math>-transversely oriented codimension one foliation <math>F</math> with a non empty set of singularities all of them centers. Then the singular set of <math>F</math> consists of two points and <math>M^n</math> is homeomorphic to the sphere <math>S^n</math>''.
+It is a consequence of the [[Reeb stability theorem]].
+==Generalization==
+More general case is <math>c>s\ge 0.</math>
+In 1978, E. Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles. He showed that the number of centers cannot be too much as compared with the number of saddles, notably, <math>c\le s+2</math>. So there are exactly two cases when <math>c>s</math>:
+:(1)   <math>c=s+2, \, </math>
+:(2)   <math>c=s+1. \, </math>
+He obtained a description of the manifold admitting a foliation with singularities that satisfy&nbsp;(1).
+Theorem:<ref>{{citation
+ | last = Wagneur | first = E.
+ | issue = 3
+ | journal = Annales de l'Institut Fourier
+ | language = French
+ | mr = 511820
+ | pages = xi, 165–176
+ | title = Formes de Pfaff à singularités non dégénérées
+ | url = http://www.numdam.org/item?id=AIF_1978__28_3_165_0
+ | volume = 28
+ | year = 1978}}.</ref> ''Let <math>M^n</math> be a compact connected manifold admitting  a Morse foliation <math>F</math> with <math>c</math> centers and <math>s</math> saddles. Then <math>c\le s+2</math>.''
+''In case <math>c=s+2</math>,''
+* ''<math>M</math> is homeomorphic to <math>S^n</math>,''
+* ''all saddles have index'' 1,
+* ''each regular leaf is diffeomorphic to <math>S^{n-1}</math>.''
+Finally, in 2008, C. Camacho and B. Scardua considered the case (2), <math>c=s+1</math>. Interestingly, this is possible in a small number of low dimensions.
+Theorem:<ref>{{citation
+ | last1 = Camacho | first1 = César
+ | last2 = Scárdua | first2 = Bruno
+ | arxiv = math/0611395
+ | doi = 10.1090/S0002-9939-08-09371-4
+ | issue = 11
+ | journal = [[Proceedings of the American Mathematical Society]]
+ | mr = 2425748
+ | pages = 4065–4073
+ | title = On foliations with Morse singularities
+ | volume = 136
+ | year = 2008}}.</ref> ''Let <math>M^n</math> be a compact connected manifold and <math>F</math> a Morse foliation on <math>M</math>. If <math>s = c + 1</math>, then''
+* ''<math>n=2,4,8</math> or <math>16</math>,''
+* ''<math>M^n</math> is an [[Eells–Kuiper manifold]].''
+==References==
+<references/>
+{{DEFAULTSORT:Reeb Sphere Theorem}}
+[[Category:Foliations]]
+[[Category:Theorems in topology]]

Zeta function (operator): Difference between revisions

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Morse foliation

Reeb sphere theorem

Generalization

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Morse foliation

Reeb sphere theorem

Generalization

References

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