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In [[differential geometry]], one can attach to every point ''x'' of a [[smooth manifold|smooth (or differentiable) manifold]] a [[vector space]] called the '''cotangent space''' at ''x''. Typically, the cotangent space is defined as the [[dual space]] of the [[tangent space]] at ''x'', although there are more direct definitions (see below). The elements of the cotangent space are called '''cotangent vectors''' or '''tangent covectors'''. | |||
==Properties== | |||
All cotangent spaces on a connected manifold have the same [[dimension of a vector space|dimension]], equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the [[cotangent bundle]] of the manifold. | |||
The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore [[isomorphic]] to each other via many possible isomorphisms. The introduction of a [[Riemannian metric]] or a [[symplectic form]] gives rise to a [[natural isomorphism]] between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector. | |||
==Formal definitions== | |||
===Definition as linear functionals=== | |||
Let ''M'' be a smooth manifold and let ''x'' be a point in ''M''. Let ''T''<sub>''x''</sub>''M'' be the [[tangent space]] at ''x''. Then the cotangent space at ''x'' is defined as the [[dual space]] of ''T''<sub>''x''</sub>''M'': | |||
:''T''<sub>''x''</sub><sup>*</sup>''M'' = (''T''<sub>''x''</sub>''M'')<sup>*</sup> | |||
Concretely, elements of the cotangent space are [[linear functional]]s on ''T''<sub>''x''</sub>''M''. That is, every element α ∈ ''T''<sub>''x''</sub><sup>*</sup>''M'' is a [[linear map]] | |||
:α : ''T''<sub>''x''</sub>''M'' → '''F''' | |||
where '''F''' is the underlying [[field (mathematics)|field]] of the vector space being considered. For example, the field of [[real number]]s. The elements of ''T''<sub>''x''</sub><sup>*</sup>''M'' are called cotangent vectors. | |||
===Alternative definition=== | |||
In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of [[equivalence class]]es of smooth functions on ''M''. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x. The cotangent space will then consist of all the possible first-order behaviors of a function near x. | |||
Let ''M'' be a smooth manifold and let ''x'' be a point in ''M''. Let ''I''<sub>''x''</sub> be the [[ideal (ring theory)|ideal]] of all functions in C<sup>∞</sup>(''M'') vanishing at ''x'', and let ''I''<sub>''x''</sub><sup>2</sup> be the set of functions of the form <math>\sum_i f_i g_i\,</math>, where ''f''<sub>''i''</sub>, ''g''<sub>''i''</sub> ∈ ''I''<sub>''x''</sub>. Then ''I''<sub>''x''</sub> and ''I''<sub>''x''</sub><sup>2</sup> are real vector spaces and the cotangent space is defined as the [[Quotient space (linear algebra)|quotient space]] ''T''<sub>''x''</sub><sup>*</sup>''M'' = ''I''<sub>''x''</sub> / ''I''<sub>''x''</sub><sup>2</sup>. | |||
This formulation is analogous to the construction of the cotangent space to define the [[Zariski tangent space]] in algebraic geometry. The construction also generalizes to [[locally ringed space]]s. | |||
==The differential of a function== | |||
Let ''M'' be a smooth manifold and let ''f'' ∈ C<sup>∞</sup>(''M'') be a [[smooth function]]. The differential of ''f'' at a point ''x'' is the map | |||
:d''f''<sub>''x''</sub>(''X''<sub>''x''</sub>) = ''X''<sub>''x''</sub>(''f'') | |||
where ''X''<sub>''x''</sub> is a [[Differential geometry of curves|tangent vector]] at ''x'', thought of as a derivation. That is <math>X(f)=\mathcal{L}_Xf</math> is the [[Lie derivative]] of ''f'' in the direction ''X'', and one has d''f''(''X'')=''X''(''f''). Equivalently, we can think of tangent vectors as tangents to curves, and write | |||
:d''f''<sub>''x''</sub>(γ′(0)) = (''f'' o γ)′(0) | |||
In either case, d''f''<sub>''x''</sub> is a linear map on ''T''<sub>''x''</sub>''M'' and hence it is a tangent covector at ''x''. | |||
We can then define the differential map d : C<sup>∞</sup>(''M'') → ''T''<sub>''x''</sub><sup>*</sup>''M'' at a point ''x'' as the map which sends ''f'' to d''f''<sub>''x''</sub>. Properties of the differential map include: | |||
# d is a linear map: d(''af'' + ''bg'') = ''a'' d''f'' + ''b'' d''g'' for constants ''a'' and ''b'', | |||
# d(''fg'')<sub>''x''</sub> = ''f''(''x'')d''g''<sub>''x''</sub> + ''g''(''x'')d''f''<sub>''x''</sub>, | |||
The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function ''f'' ∈ ''I''<sub>''x''</sub> (a smooth function vanishing at ''x'') we can form the linear functional d''f''<sub>''x''</sub> as above. Since the map d restricts to 0 on ''I''<sub>''x''</sub><sup>2</sup> (the reader should verify this), d descends to a map from ''I''<sub>''x''</sub> / ''I''<sub>''x''</sub><sup>2</sup> to the dual of the tangent space, (''T''<sub>''x''</sub>''M'')<sup>*</sup>. One can show that this map is an isomorphism, establishing the equivalence of the two definitions. | |||
==The pullback of a smooth map== | |||
Just as every differentiable map ''f'' : ''M'' → ''N'' between manifolds induces a linear map (called the ''pushforward'' or ''derivative'') between the tangent spaces | |||
:<math>f_{*}^{}\colon T_x M \to T_{f(x)} N</math> | |||
every such map induces a linear map (called the ''[[pullback (differential geometry)|pullback]]'') between the cotangent spaces, only this time in the reverse direction: | |||
:<math>f^{*}\colon T_{f(x)}^{*} N \to T_{x}^{*} M</math> | |||
The pullback is naturally defined as the dual (or transpose) of the [[pushforward (differential)|pushforward]]. Unraveling the definition, this means the following: | |||
:<math>(f^{*}\theta)(X_x) = \theta(f_{*}^{}X_x)</math> | |||
where θ ∈ ''T''<sub>''f''(''x'')</sub><sup>*</sup>''N'' and ''X''<sub>''x''</sub> ∈ ''T''<sub>''x''</sub>''M''. Note carefully where everything lives. | |||
If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let ''g'' be a smooth function on ''N'' vanishing at ''f''(''x''). Then the pullback of the covector determined by ''g'' (denoted d''g'') is given by | |||
:<math>f^{*}\mathrm dg = \mathrm d(g \circ f).</math> | |||
That is, it is the equivalence class of functions on ''M'' vanishing at ''x'' determined by ''g'' o ''f''. | |||
==Exterior powers== | |||
The ''k''-th [[exterior power]] of the cotangent space, denoted Λ<sup>''k''</sup>(''T''<sub>''x''</sub><sup>*</sup>''M''), is another important object in differential geometry. Vectors in the ''k''th exterior power, or more precisely sections of the ''k''-th exterior power of the [[cotangent bundle]], are called [[differential form|differential ''k''-forms]]. They can be thought of as alternating, [[multilinear map]]s on ''k'' tangent vectors. | |||
For this reason, tangent covectors are frequently called ''[[one-form]]s''. | |||
==References== | |||
* {{Citation | last1=Abraham | first1=Ralph H. | author1-link=Ralph Abraham | last2=Marsden | first2=Jerrold E. | author2-link=Jerrold E. Marsden | title=Foundations of mechanics | publisher=Benjamin-Cummings | location=London | isbn=978-0-8053-0102-1 | year=1978}} | |||
* {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=4th | isbn=978-3-540-25907-7 | year=2005}} | |||
* {{Citation | last1=Lee | first1=John M. | title=Introduction to smooth manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Graduate Texts in Mathematics | isbn=978-0-387-95448-6 | year=2003 | volume=218}} | |||
* {{Citation | last1=Misner | first1=Charles W. | author1-link=Charles W. Misner | last2=Thorne | first2=Kip | author2-link=Kip Thorne | last3=Wheeler | first3=John Archibald | author3-link=John Archibald Wheeler | title=[[Gravitation (book)|Gravitation]] | publisher=W. H. Freeman | isbn=978-0-7167-0344-0 | year=1973}} | |||
{{DEFAULTSORT:Cotangent Space}} | |||
[[Category:Differential topology]] | |||
Revision as of 09:09, 9 March 2013
In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors.
Properties
All cotangent spaces on a connected manifold have the same dimension, equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold.
The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms. The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.
Formal definitions
Definition as linear functionals
Let M be a smooth manifold and let x be a point in M. Let TxM be the tangent space at x. Then the cotangent space at x is defined as the dual space of TxM:
- Tx*M = (TxM)*
Concretely, elements of the cotangent space are linear functionals on TxM. That is, every element α ∈ Tx*M is a linear map
- α : TxM → F
where F is the underlying field of the vector space being considered. For example, the field of real numbers. The elements of Tx*M are called cotangent vectors.
Alternative definition
In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x. The cotangent space will then consist of all the possible first-order behaviors of a function near x.
Let M be a smooth manifold and let x be a point in M. Let Ix be the ideal of all functions in C∞(M) vanishing at x, and let Ix2 be the set of functions of the form , where fi, gi ∈ Ix. Then Ix and Ix2 are real vector spaces and the cotangent space is defined as the quotient space Tx*M = Ix / Ix2.
This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry. The construction also generalizes to locally ringed spaces.
The differential of a function
Let M be a smooth manifold and let f ∈ C∞(M) be a smooth function. The differential of f at a point x is the map
- dfx(Xx) = Xx(f)
where Xx is a tangent vector at x, thought of as a derivation. That is is the Lie derivative of f in the direction X, and one has df(X)=X(f). Equivalently, we can think of tangent vectors as tangents to curves, and write
- dfx(γ′(0)) = (f o γ)′(0)
In either case, dfx is a linear map on TxM and hence it is a tangent covector at x.
We can then define the differential map d : C∞(M) → Tx*M at a point x as the map which sends f to dfx. Properties of the differential map include:
- d is a linear map: d(af + bg) = a df + b dg for constants a and b,
- d(fg)x = f(x)dgx + g(x)dfx,
The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function f ∈ Ix (a smooth function vanishing at x) we can form the linear functional dfx as above. Since the map d restricts to 0 on Ix2 (the reader should verify this), d descends to a map from Ix / Ix2 to the dual of the tangent space, (TxM)*. One can show that this map is an isomorphism, establishing the equivalence of the two definitions.
The pullback of a smooth map
Just as every differentiable map f : M → N between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces
every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction:
The pullback is naturally defined as the dual (or transpose) of the pushforward. Unraveling the definition, this means the following:
where θ ∈ Tf(x)*N and Xx ∈ TxM. Note carefully where everything lives.
If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let g be a smooth function on N vanishing at f(x). Then the pullback of the covector determined by g (denoted dg) is given by
That is, it is the equivalence class of functions on M vanishing at x determined by g o f.
Exterior powers
The k-th exterior power of the cotangent space, denoted Λk(Tx*M), is another important object in differential geometry. Vectors in the kth exterior power, or more precisely sections of the k-th exterior power of the cotangent bundle, are called differential k-forms. They can be thought of as alternating, multilinear maps on k tangent vectors. For this reason, tangent covectors are frequently called one-forms.
References
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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