Oja's rule

In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds. They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.

In the classical differential geometry of surfaces, the Gauss–Codazzi–Mainardi equations consist of a pair of related equations. The first equation, sometimes called the Gauss equation, relates the intrinsic curvature (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form. This equation is the basis for Gauss's theorema egregium. The second equation, sometimes called the Codazzi–Mainardi equation, is a structural condition on the second derivatives of the Gauss map. It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by Template:Harvtxt. It incorporates the extrinsic curvature (or mean curvature) of the surface. The equations show that the components of the second fundamental form and its derivatives along the surface completely classify the surface up to a Euclidean transformation, a theorem of Ossian Bonnet.

Formal statement

Let i : M ⊂ P be an n-dimensional embedded submanifold of a Riemannian manifold P of dimension n+p. There is a natural inclusion of the tangent bundle of M into that of P by the pushforward, and the cokernel is the normal bundle of M:

$0\to T_{x}M\to T_{x}P{|}_M\to T_x^{\perp }M\to 0.$

The metric splits this short exact sequence, and so

$TP{|}_M=TM\oplus T^{\perp }M.$

Relative to this splitting, the Levi-Civita connection ∇′ of P decomposes into tangential and normal components. For each X ∈ TM and vector field Y on M,

$\mathrm{\nabla }{\text{'}}_{X}Y=\mathrm{\top }(\mathrm{\nabla }{\text{'}}_{X}Y)+\mathrm{\perp }(\mathrm{\nabla }{\text{'}}_{X}Y).$

Let

${\mathrm{\nabla }}_{X}Y=\mathrm{\top }(\mathrm{\nabla }{\text{'}}_{X}Y),\alpha (X,Y)=\mathrm{\perp }(\mathrm{\nabla }{\text{'}}_{X}Y).$

Gauss' formula now asserts that ∇_X is the Levi-Civita connection for M, and α is a symmetric vector-valued form with values in the normal bundle. It is often referred to as the second fundamental form.

An immediate corollary is the Gauss equation. For X, Y, Z, W ∈ TM,

$⟨R^{\prime }(X,Y)Z,W⟩=⟨R(X,Y)Z,W⟩+⟨\alpha (X,Z),\alpha (Y,W)⟩-⟨\alpha (Y,Z),\alpha (X,W)⟩$

where R′ is the Riemann curvature tensor of P and R is that of M.

The Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let X ∈ TM and ξ a normal vector field. Then decompose the ambient covariant derivative of ξ along X into tangential and normal components:

$\mathrm{\nabla }{\text{'}}_X\xi =\mathrm{\top }(\mathrm{\nabla }{\text{'}}_X\xi )+\mathrm{\perp }(\mathrm{\nabla }{\text{'}}_X\xi )=-A_{\xi }(X)+D_X(\xi ).$

Then

1. Weingarten's equation: $⟨A_{\xi }X,Y⟩=⟨\alpha (X,Y),\xi ⟩$
2. D_X is a metric connection in the normal bundle.

There are thus a pair of connections: ∇, defined on the tangent bundle of M; and D, defined on the normal bundle of M. These combine to form a connection on any tensor product of copies of TM and T^⊥M. In particular, they defined the covariant derivative of α:

$({\tilde{\mathrm{\nabla }}}_X\alpha )(Y,Z)=D_X(\alpha (Y,Z))-\alpha ({\mathrm{\nabla }}_{X}Y,Z)-\alpha (Y,{\mathrm{\nabla }}_{X}Z).$

The Codazzi–Mainardi equation is

$\mathrm{\perp }(R^{\prime }(X,Y)Z)=({\tilde{\mathrm{\nabla }}}_X\alpha )(Y,Z)-({\tilde{\mathrm{\nabla }}}_Y\alpha )(X,Z).$

Since every immersion is, in particular, a local embedding, the above formulas also hold for immersions.

Gauss–Codazzi equations in classical differential geometry

Statement of classical equations

In classical differential geometry of surfaces, the Codazzi–Mainardi equations are expressed via the second fundamental form (L, M, N):

$L_v-M_u=L{\mathrm{\Gamma }}^1{}_{12}+M({\mathrm{\Gamma }}^2{}_{12}-{\mathrm{\Gamma }}^1{}_{11})-N{\mathrm{\Gamma }}^2{}_{11}$

$M_v-N_u=L{\mathrm{\Gamma }}^1{}_{22}+M({\mathrm{\Gamma }}^2{}_{22}-{\mathrm{\Gamma }}^1{}_{12})-N{\mathrm{\Gamma }}^2{}_{12}$

Derivation of classical equations

Consider a parametric surface in Euclidean space,

$\mathbf{r}(u,v)=(x(u,v),y(u,v),z(u,v))$

where the three component functions depend smoothly on ordered pairs (u,v) in some open domain U in the uv-plane. Assume that this surface is regular, meaning that the vectors r_u and r_v are linearly independent. Complete this to a basis{r_u,r_v,n}, by selecting a unit vector n normal to the surface. It is possible to express the second partial derivatives of r using the Christoffel symbols and the second fundamental form.

${\mathbf{r}}_{uu}={\mathrm{\Gamma }}^1{}_{11}{\mathbf{r}}_u+{\mathrm{\Gamma }}^2{}_{11}{\mathbf{r}}_v+L\mathbf{n}$

${\mathbf{r}}_{uv}={\mathrm{\Gamma }}^1{}_{12}{\mathbf{r}}_u+{\mathrm{\Gamma }}^2{}_{12}{\mathbf{r}}_v+M\mathbf{n}$

${\mathbf{r}}_{vv}={\mathrm{\Gamma }}^1{}_{22}{\mathbf{r}}_u+{\mathrm{\Gamma }}^2{}_{22}{\mathbf{r}}_v+N\mathbf{n}$

Clairaut's theorem states that partial derivatives commute:

${\left({\mathbf{r}}_{uu}\right)}_v={\left({\mathbf{r}}_{uv}\right)}_u$

If we differentiate r_uu with respect to v and r_uv with respect to u, we get:

${\left({\mathrm{\Gamma }}^1{}_{11}\right)}_v{\mathbf{r}}_u+{\mathrm{\Gamma }}^1{}_{11}{\mathbf{r}}_{uv}+{\left({\mathrm{\Gamma }}^2{}_{11}\right)}_v{\mathbf{r}}_v+{\mathrm{\Gamma }}^2{}_{11}{\mathbf{r}}_{vv}+L_v\mathbf{n}+L{\mathbf{n}}_v$ $={\left({\mathrm{\Gamma }}^1{}_{12}\right)}_u{\mathbf{r}}_u+{\mathrm{\Gamma }}^1{}_{12}{\mathbf{r}}_{uu}+{\left({\mathrm{\Gamma }}_{12}^2\right)}_u{\mathbf{r}}_v+{\mathrm{\Gamma }}^2{}_{12}{\mathbf{r}}_{uv}+M_u\mathbf{n}+M{\mathbf{n}}_u$

Now substitute the above expressions for the second derivatives and equate the coefficients of n:

$M{\mathrm{\Gamma }}^1{}_{11}+N{\mathrm{\Gamma }}^2{}_{11}+L_v=L{\mathrm{\Gamma }}^1{}_{12}+M{\mathrm{\Gamma }}^2{}_{12}+M_u$

Rearranging this equation gives the first Codazzi–Mainardi equation.

The second equation may be derived similarly.

Mean curvature

Let M be a smooth m-dimensional manifold immersed in the (m + k)-dimensional smooth manifold P. Let $e_1,e_2,\dots ,e_k$ be a local orthonormal frame of vector fields normal to M. Then we can write,

$\alpha (X,Y)=\sum _{j=1}^k{\alpha }_j(X,Y)e_j$

If, now, $E_1,E_2,\dots ,E_m$ is a local orthonormal frame (of tangent vector fields) on the same open subset of M, then we can define the mean curvatures of the immersion by

$H_j=\sum _{i=1}^m{\alpha }_j(E_i,E_i)$

In particular, if M is a hypersurface of P, i.e. $k=1$, then there is only one mean curvature to speak of. The immersion is called minimal if all the $H_j$ are identically zero.

Observe that the mean curvature is a trace, or average, of the second fundamental form, for any given component. Sometimes mean curvature is defined by multiplying the sum on the right-hand side by $1/m$.

We can now write the Gauss–Codazzi equations as

$⟨R^{\prime }(X,Y)Z,W⟩=⟨R(X,Y)Z,W⟩+\sum _{j=1}^k{\alpha }_j(X,Z){\alpha }_j(Y,W)-{\alpha }_j(Y,Z){\alpha }_j(X,W)$

Contracting the $Y,Z$ components gives us

$Ri{c}^{\prime }(X,W)=Ric(X,W)+\sum _{j=1}^k⟨R^{\prime }(X,e_j)e_j,W⟩+(\sum _{i=1}^m{\alpha }_j(X,E_i){\alpha }_j(E_i,W))-H_j{\alpha }_j(X,W)$

Observe that the tensor in parentheses is symmetric and nonnegative-definite in $X,W$. Assuming that M is a hypersurface, this simplifies to

$Ri{c}^{\prime }(X,W)=Ric(X,W)+⟨R^{\prime }(X,n)n,W⟩+(\sum _{i=1}^{m}h(X,E_i)h(E_i,W))-Hh(X,W)$

where $n=e_1$ and $h={\alpha }_1$ and $H=H_1$. In that case, one more contraction yields,

$R^{\prime }=R+2Ri{c}^{\prime }(n,n)+\Vert h{\Vert }^2-H^2$

where $R^{\prime }$ and $R$ are the respective scalar curvatures, and

$\Vert h{\Vert }^2=\sum _{i,j=1}^{m}h(E_i,E_j{)}^2$

If $k>1$, the scalar curvature equation might be more complicated.

We can already use these equations to draw some conclusions. For example, any minimal immersion into the round sphere $x_1^2+x_2^2+\cdots +x_{m+k+1}^2=1$ must be of the form

$\mathrm{△}{x}_j+\lambda x_j=0$

where $j$ runs from 1 to $m+k+1$ and

$\mathrm{△}=\sum _{i=1}^m{\mathrm{\nabla }}_{E_i}{\mathrm{\nabla }}_{E_i}$

is the Laplacian on M, and $\lambda >0$ is a positive constant.

See also

• Darboux frame

Notes

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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.
  
  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.
  
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  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 ("General Discussions about Curved Surfaces")
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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.
  
  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.
• 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

External links

• Peterson–Mainardi–Codazzi Equations – from Wolfram MathWorld
• Peterson–Codazzi Equations