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83-458: A Hermitian form in (the vector space) C defines a unitary subgroup U( n +1) in GL( n +1, C ). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U( n +1) action; thus it is homogeneous . Equipped with a Fubini–Study metric, CP is a symmetric space . The particular normalization on the metric depends on the application. In Riemannian geometry , one uses

166-419: A {\displaystyle \omega _{\;\;b}^{a}} that satisfies the torsion-free condition and is covariantly constant, which, for spin connections, means that it is antisymmetric in the vierbein indexes: The above is readily solved; one obtains The curvature 2-form is defined as Hermitian form In mathematics , a sesquilinear form is a generalization of a bilinear form that, in turn,

249-566: A , b ∈ C . {\displaystyle a,b\in \mathbb {C} .} Here, a ¯ {\displaystyle {\overline {a}}} is the complex conjugate of a scalar a . {\displaystyle a.} A complex sesquilinear form can also be viewed as a complex bilinear map V ¯ × V → C {\displaystyle {\overline {V}}\times V\to \mathbb {C} } where V ¯ {\displaystyle {\overline {V}}}

332-401: A complex differential form ω of degree (1,1). The form ω is defined as minus the imaginary part of h : ω = i 2 ( h − h ¯ ) . {\displaystyle \omega ={i \over 2}\left(h-{\bar {h}}\right).} Again since ω is equal to its conjugate it is the complexification of a real form on TM . The form ω

415-414: A division ring , Reinhold Baer extended the definition of a sesquilinear form to a division ring, which requires replacing vector spaces by R -modules . (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.) The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to

498-405: A prime power . With respect to the standard basis we can write x = ( x 1 , x 2 , x 3 ) and y = ( y 1 , y 2 , y 3 ) and define the map φ by: The map σ  : t ↦ t is an involutory automorphism of F . The map φ is then a σ -sesquilinear form. The matrix M φ associated to this form is the identity matrix . This is a Hermitian form. In

581-564: A projective geometry G , a permutation δ of the subspaces that inverts inclusion, i.e. is called a correlation . A result of Birkhoff and von Neumann (1936) shows that the correlations of desarguesian projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space. A sesquilinear form φ is nondegenerate if φ ( x , y ) = 0 for all y in V (if and) only if x = 0 . To achieve full generality of this statement, and since every desarguesian projective geometry may be coordinatized by

664-456: A Hermitian form. The matrix representation of a complex skew-Hermitian form is a skew-Hermitian matrix . A complex skew-Hermitian form applied to a single vector | z | s = s ( z , z ) {\displaystyle |z|_{s}=s(z,z)} is always a purely imaginary number . This section applies unchanged when the division ring K is commutative . More specific terminology then also applies:

747-428: A Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space . One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure . A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields a unitary structure ( U(n) structure ) on

830-452: A Hermitian metric on an almost complex manifold M is equivalent to a choice of U( n )-structure on M ; that is, a reduction of the structure group of the frame bundle of M from GL( n , C ) to the unitary group U( n ). A unitary frame on an almost Hermitian manifold is complex linear frame which is orthonormal with respect to the Hermitian metric. The unitary frame bundle of M

913-479: A broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector. A motivating special case is a sesquilinear form on a complex vector space , V . This is a map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows

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996-539: A complex Hermitian form is a Hermitian matrix . A complex Hermitian form applied to a single vector | z | h = h ( z , z ) {\displaystyle |z|_{h}=h(z,z)} is always a real number . One can show that a complex sesquilinear form is Hermitian if and only if the associated quadratic form is real for all z ∈ V . {\displaystyle z\in V.} A complex skew-Hermitian form (also called an antisymmetric sesquilinear form ),

1079-421: A counterclockwise rotation about the origin by an angle θ {\displaystyle \theta } , the quotient mapping C  →  CP splits into two pieces. where step (a) is a quotient by the dilation Z  ~  R Z for R  ∈  R , the multiplicative group of positive real numbers , and step (b) is a quotient by the rotations Z  ~  e Z . The result of

1162-540: A frame { ∂ 1 , … , ∂ n } {\displaystyle \{\partial _{1},\ldots ,\partial _{n}\}} of the holomorphic tangent bundle of CP , in terms of which the Fubini–Study metric has Hermitian components where | z | = | z 1 | + ... + | z n |. That is, the Hermitian matrix of the Fubini–Study metric in this frame

1245-413: A generalized concept of "complex conjugation" for the ring. Conventions differ as to which argument should be linear. In the commutative case, we shall take the first to be linear, as is common in the mathematical literature, except in the section devoted to sesquilinear forms on complex vector spaces. There we use the other convention and take the first argument to be conjugate-linear (i.e. antilinear) and

1328-501: A hermitian metric on a holomorphic vector bundle . The most important class of Hermitian manifolds are Kähler manifolds . These are Hermitian manifolds for which the Hermitian form ω is closed : d ω = 0 . {\displaystyle d\omega =0\,.} In this case the form ω is called a Kähler form . A Kähler form is a symplectic form , and so Kähler manifolds are naturally symplectic manifolds . An almost Hermitian manifold whose associated (1,1)-form

1411-403: A normalization so that the Fubini–Study metric simply relates to the standard metric on the (2 n +1)-sphere . In algebraic geometry , one uses a normalization making CP a Hodge manifold . The Fubini–Study metric arises naturally in the quotient space construction of complex projective space . Specifically, one may define CP to be the space consisting of all complex lines in C , i.e.,

1494-407: A quotient is taken of a Riemannian manifold (or metric space in general), care must be taken to ensure that the quotient space is endowed with a metric that is well-defined. For instance, if a group G acts on a Riemannian manifold ( X , g ), then in order for the orbit space X / G to possess an induced metric, g {\displaystyle g} must be constant along G -orbits in

1577-401: A right (left) R -module V can be turned into a left (right) R -module, V . Thus, the sesquilinear form φ  : V × V → R can be viewed as a bilinear form φ ′ : V × V → R . Hermitian metric In mathematics , and more specifically in differential geometry , a Hermitian manifold is the complex analogue of a Riemannian manifold . More precisely,

1660-414: A routine computation shows where d s u s 2 {\displaystyle ds_{us}^{2}} is the round metric on the unit 2-sphere. Here φ, θ are "mathematician's spherical coordinates " on S coming from the stereographic projection r  tan(φ/2) = 1, tan θ =  y / x . (Many physics references interchange the roles of φ and θ.) The Kähler form

1743-828: A sesquilinear form is represented by a matrix A , {\displaystyle A,} and given by φ ( w , z ) = φ ( ∑ i w i e i , ∑ j z j e j ) = ∑ i ∑ j w i ¯ z j φ ( e i , e j ) = w † A z . {\displaystyle \varphi (w,z)=\varphi \left(\sum _{i}w_{i}e_{i},\sum _{j}z_{j}e_{j}\right)=\sum _{i}\sum _{j}{\overline {w_{i}}}z_{j}\varphi \left(e_{i},e_{j}\right)=w^{\dagger }Az.} where w † {\displaystyle w^{\dagger }}

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1826-514: Is Choosing as vierbeins e 1 = d x / ( 1 + r 2 ) {\displaystyle e^{1}=dx/(1+r^{2})} and e 2 = d y / ( 1 + r 2 ) {\displaystyle e^{2}=dy/(1+r^{2})} , the Kähler form simplifies to Applying the Hodge star to the Kähler form, one obtains implying that K

1909-460: Is Note that each matrix element is unitary-invariant: the diagonal action z ↦ e i θ z {\displaystyle \mathbf {z} \mapsto e^{i\theta }\mathbf {z} } will leave this matrix unchanged. Accordingly, the line element is given by In this last expression, the summation convention is used to sum over Latin indices i , j that range from 1 to  n . The metric can be derived from

1992-399: Is ( σ , ε ) -Hermitian for some ε . In the special case that σ is the identity map (i.e., σ = id ), K is commutative, φ is a bilinear form and ε = 1 . Then for ε = 1 the bilinear form is called symmetric , and for ε = −1 is called skew-symmetric . Let V be the three dimensional vector space over the finite field F = GF( q ) , where q is

2075-440: Is Hermitian if there exists σ such that for all x , y in V . A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism σ is an involution (i.e. of order 2). Since for an antiautomorphism σ we have σ ( st ) = σ ( t ) σ ( s ) for all s , t in R , if σ = id , then R must be commutative and φ is a bilinear form. In particular, if, in this case, R

2158-477: Is harmonic . The Fubini–Study metric on the complex projective plane CP has been proposed as a gravitational instanton , the gravitational analog of an instanton . The metric, the connection form and the curvature are readily computed, once suitable real 4D coordinates are established. Writing ( x , y , z , t ) {\displaystyle (x,y,z,t)} for real Cartesian coordinates, one then defines polar coordinate one-forms on

2241-430: Is orthogonal to another element y with respect to the sesquilinear form φ (written x ⊥ y ) if φ ( x , y ) = 0 . This relation need not be symmetric, i.e. x ⊥ y does not imply y ⊥ x . A sesquilinear form φ  : V × V → R is reflexive (or orthosymmetric ) if φ ( x , y ) = 0 implies φ ( y , x ) = 0 for all x , y in V . A sesquilinear form φ  : V × V → R

2324-416: Is reflexive if, for all x , y in M , That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric. A σ -sesquilinear form φ is called ( σ , ε ) -Hermitian if there exists ε in K such that, for all x , y in M , If ε = 1 , the form is called σ - Hermitian , and if ε = −1 , it is called σ - anti-Hermitian . (When σ

2407-818: Is a complex manifold with a Hermitian metric on its holomorphic tangent bundle . Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent bundle. On a Hermitian manifold the metric can be written in local holomorphic coordinates ( z α ) {\displaystyle (z^{\alpha })} as h = h α β ¯ d z α ⊗ d z ¯ β {\displaystyle h=h_{\alpha {\bar {\beta }}}\,dz^{\alpha }\otimes d{\bar {z}}^{\beta }} where h α β ¯ {\displaystyle h_{\alpha {\bar {\beta }}}} are

2490-732: Is a conjugate-linear functional on V . {\displaystyle V.} Given any complex sesquilinear form φ {\displaystyle \varphi } on V {\displaystyle V} we can define a second complex sesquilinear form ψ {\displaystyle \psi } via the conjugate transpose : ψ ( w , z ) = φ ( z , w ) ¯ . {\displaystyle \psi (w,z)={\overline {\varphi (z,w)}}.} In general, ψ {\displaystyle \psi } and φ {\displaystyle \varphi } will be different. If they are

2573-401: Is a Hermitian form. A minus sign is introduced in the Hermitian form w w ∗ − z z ∗ {\displaystyle ww^{*}-zz^{*}} to define the group SU(1,1) . A vector space with a Hermitian form ( V , h ) {\displaystyle (V,h)} is called a Hermitian space . The matrix representation of

Fubini–Study metric - Misplaced Pages Continue

2656-509: Is a complex sesquilinear form s : V × V → C {\displaystyle s:V\times V\to \mathbb {C} } such that s ( w , z ) = − s ( z , w ) ¯ . {\displaystyle s(w,z)=-{\overline {s(z,w)}}.} Every complex skew-Hermitian form can be written as the imaginary unit i := − 1 {\displaystyle i:={\sqrt {-1}}} times

2739-493: Is a generalization of the concept of the dot product of Euclidean space . A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing

2822-445: Is a skewfield, then R is a field and V is a vector space with a bilinear form. An antiautomorphism σ  : R → R can also be viewed as an isomorphism R → R , where R is the opposite ring of R , which has the same underlying set and the same addition, but whose multiplication operation ( ∗ ) is defined by a ∗ b = ba , where the product on the right is the product in R . It follows from this that

2905-854: Is a symmetric bilinear form on TM , the complexified tangent bundle. Since g is equal to its conjugate it is the complexification of a real form on TM . The symmetry and positive-definiteness of g on TM follow from the corresponding properties of h . In local holomorphic coordinates the metric g can be written g = 1 2 h α β ¯ ( d z α ⊗ d z ¯ β + d z ¯ β ⊗ d z α ) . {\displaystyle g={1 \over 2}h_{\alpha {\bar {\beta }}}\,\left(dz^{\alpha }\otimes d{\bar {z}}^{\beta }+d{\bar {z}}^{\beta }\otimes dz^{\alpha }\right).} One can also associate to h

2988-485: Is called variously the associated (1,1) form , the fundamental form , or the Hermitian form . In local holomorphic coordinates ω can be written ω = i 2 h α β ¯ d z α ∧ d z ¯ β . {\displaystyle \omega ={i \over 2}h_{\alpha {\bar {\beta }}}\,dz^{\alpha }\wedge d{\bar {z}}^{\beta }.} It

3071-688: Is clear from the coordinate representations that any one of the three forms h , g , and ω uniquely determine the other two. The Riemannian metric g and associated (1,1) form ω are related by the almost complex structure J as follows ω ( u , v ) = g ( J u , v ) g ( u , v ) = ω ( u , J v ) {\displaystyle {\begin{aligned}\omega (u,v)&=g(Ju,v)\\g(u,v)&=\omega (u,Jv)\end{aligned}}} for all complex tangent vectors u and v . The Hermitian metric h can be recovered from g and ω via

3154-408: Is closed is naturally called an almost Kähler manifold . Any symplectic manifold admits a compatible almost complex structure making it into an almost Kähler manifold. A Kähler manifold is an almost Hermitian manifold satisfying an integrability condition . This can be stated in several equivalent ways. Let ( M , g , ω, J ) be an almost Hermitian manifold of real dimension 2 n and let ∇ be

3237-442: Is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by ⟨ w , z ⟩ = ∑ i = 1 n w ¯ i z i . {\displaystyle \langle w,z\rangle =\sum _{i=1}^{n}{\overline {w}}_{i}z_{i}.} More generally, the inner product on any complex Hilbert space

3320-472: Is given by The vierbeins can be immediately read off from the last expression: That is, in the vierbein coordinate system, using roman-letter subscripts, the metric tensor is Euclidean: Given the vierbein, a spin connection can be computed; the Levi-Civita spin connection is the unique connection that is torsion-free and covariantly constant, namely, it is the one-form ω b

3403-412: Is implied, respectively simply Hermitian or anti-Hermitian .) For a nonzero ( σ , ε ) -Hermitian form, it follows that for all α in K , It also follows that φ ( x , x ) is a fixed point of the map α ↦ σ ( α ) ε . The fixed points of this map form a subgroup of the additive group of K . A ( σ , ε ) -Hermitian form is reflexive, and every reflexive σ -sesquilinear form

Fubini–Study metric - Misplaced Pages Continue

3486-426: Is not working with an orthonormal basis for C , or even any basis at all. By inserting an extra factor of i {\displaystyle i} into the product, one obtains the skew-Hermitian form , defined more precisely, below. There is no particular reason to restrict the definition to the complex numbers; it can be defined for arbitrary rings carrying an antiautomorphism , informally understood to be

3569-612: Is occasionally called the quantum angle . The angle is real-valued, and runs from 0 to π / 2 {\displaystyle \pi /2} . The infinitesimal form of this metric may be quickly obtained by taking φ = ψ + δ ψ {\displaystyle \varphi =\psi +\delta \psi } , or equivalently, W α = Z α + d Z α {\displaystyle W_{\alpha }=Z_{\alpha }+dZ_{\alpha }} to obtain In

3652-477: Is the complex conjugate vector space to V . {\displaystyle V.} By the universal property of tensor products these are in one-to-one correspondence with complex linear maps V ¯ ⊗ V → C . {\displaystyle {\overline {V}}\otimes V\to \mathbb {C} .} For a fixed z ∈ V {\displaystyle z\in V}

3735-798: Is the conjugate transpose . The components of the matrix A {\displaystyle A} are given by A i j := φ ( e i , e j ) . {\displaystyle A_{ij}:=\varphi \left(e_{i},e_{j}\right).} A complex Hermitian form (also called a symmetric sesquilinear form ), is a sesquilinear form h : V × V → C {\displaystyle h:V\times V\to \mathbb {C} } such that h ( w , z ) = h ( z , w ) ¯ . {\displaystyle h(w,z)={\overline {h(z,w)}}.} The standard Hermitian form on C n {\displaystyle \mathbb {C} ^{n}}

3818-565: Is the principal U( n )-bundle of all unitary frames. Every almost Hermitian manifold M has a canonical volume form which is just the Riemannian volume form determined by g . This form is given in terms of the associated (1,1)-form ω by v o l M = ω n n ! ∈ Ω n , n ( M ) {\displaystyle \mathrm {vol} _{M}={\frac {\omega ^{n}}{n!}}\in \Omega ^{n,n}(M)} where ω

3901-792: Is the wedge product of ω with itself n times. The volume form is therefore a real ( n , n )-form on M . In local holomorphic coordinates the volume form is given by v o l M = ( i 2 ) n det ( h α β ¯ ) d z 1 ∧ d z ¯ 1 ∧ ⋯ ∧ d z n ∧ d z ¯ n . {\displaystyle \mathrm {vol} _{M}=\left({\frac {i}{2}}\right)^{n}\det \left(h_{\alpha {\bar {\beta }}}\right)\,dz^{1}\wedge d{\bar {z}}^{1}\wedge \dotsb \wedge dz^{n}\wedge d{\bar {z}}^{n}.} One can also consider

3984-407: Is the standard notation for a point in the projective space CP in homogeneous coordinates . Then, given two points | ψ ⟩ = Z α {\displaystyle \vert \psi \rangle =Z_{\alpha }} and | φ ⟩ = W α {\displaystyle \vert \varphi \rangle =W_{\alpha }} in the space,

4067-425: Is thus identified with an equivalence class of ( n +1)-tuples [ Z 0 ,..., Z n ] modulo nonzero complex rescaling; the Z i are called homogeneous coordinates of the point. Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar z  =  R   e can be uniquely thought of as the composition of a dilation by the modulus R followed by

4150-544: Is uniquely determined by φ . Given a sesquilinear form φ over a module M and a subspace ( submodule ) W of M , the orthogonal complement of W with respect to φ is Similarly, x ∈ M is orthogonal to y ∈ M with respect to φ , written x ⊥ φ y (or simply x ⊥ y if φ can be inferred from the context), when φ ( x , y ) = 0 . This relation need not be symmetric , i.e. x ⊥ y does not imply y ⊥ x (but see § Reflexivity below). A sesquilinear form φ

4233-1046: The 4-sphere (the quaternionic projective line ) as The σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} are the standard left-invariant one-form coordinate frame on the Lie group S U ( 2 ) = S 3 {\displaystyle SU(2)=S^{3}} ; that is, they obey d σ i = 2 σ j ∧ σ k {\displaystyle d\sigma _{i}=2\sigma _{j}\wedge \sigma _{k}} for i , j , k = 1 , 2 , 3 {\displaystyle i,j,k=1,2,3} and cyclic permutations. The corresponding local affine coordinates are z 1 = x + i y {\displaystyle z_{1}=x+iy} and z 2 = z + i t {\displaystyle z_{2}=z+it} then provide with

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4316-558: The "special" Hopf fibration S  →  S  →  S . When the Fubini–Study metric is written in coordinates on CP , its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (and Gaussian curvature 4) on S . Namely, if z  =  x  + i y is the standard affine coordinate chart on the Riemann sphere CP and x  =  r  cos θ, y  =  r  sin θ are polar coordinates on C , then

4399-609: The Hermitian metric. Every (almost) complex manifold admits a Hermitian metric. This follows directly from the analogous statement for Riemannian metric. Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner: g ′ ( u , v ) = 1 2 ( g ( u , v ) + g ( J u , J v ) ) . {\displaystyle g'(u,v)={1 \over 2}\left(g(u,v)+g(Ju,Jv)\right).} Choosing

4482-633: The Kähler scalar) of CP . In quantum mechanics , the Fubini–Study metric is also known as the Bures metric . However, the Bures metric is typically defined in the notation of mixed states , whereas the exposition below is written in terms of a pure state . The real part of the metric is (a quarter of) the Fisher information metric . The Fubini–Study metric may be written using the bra–ket notation commonly used in quantum mechanics . To explicitly equate this notation to

4565-451: The almost complex structure and the fundamental form are integrable, then we have a Kähler structure . A Hermitian metric on a complex vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be viewed as a smooth global section h {\displaystyle h} of

4648-509: The basic notion of a Hermitian form on complex vector space . Hermitian forms are commonly seen in physics , as the inner product on a complex Hilbert space . In such cases, the standard Hermitian form on C is given by where w ¯ i {\displaystyle {\overline {w}}_{i}} denotes the complex conjugate of w i   . {\displaystyle w_{i}~.} This product may be generalized to situations where one

4731-518: The choice of section: this can be done by a direct calculation. The Kähler form of this metric is where the ∂ , ∂ ¯ {\displaystyle \partial ,{\bar {\partial }}} are the Dolbeault operators . The pullback of this is clearly independent of the choice of holomorphic section. The quantity log| Z | is the Kähler potential (sometimes called

4814-424: The components of a positive-definite Hermitian matrix . A Hermitian metric h on an (almost) complex manifold M defines a Riemannian metric g on the underlying smooth manifold. The metric g is defined to be the real part of h : g = 1 2 ( h + h ¯ ) . {\displaystyle g={1 \over 2}\left(h+{\bar {h}}\right).} The form g

4897-659: The context of quantum mechanics , CP is called the Bloch sphere ; the Fubini–Study metric is the natural metric for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, including quantum entanglement and the Berry phase effect, can be attributed to the peculiarities of the Fubini–Study metric. When n = 1, there is a diffeomorphism S 2 ≅ C P 1 {\displaystyle S^{2}\cong \mathbf {CP} ^{1}} given by stereographic projection . This leads to

4980-448: The coordinate patch U 0 = { Z 0 ≠ 0 } {\displaystyle U_{0}=\{Z_{0}\neq 0\}} . One can develop an affine coordinate system in any of the coordinate patches U i = { Z i ≠ 0 } {\displaystyle U_{i}=\{Z_{i}\neq 0\}} by dividing instead by Z i {\displaystyle Z_{i}} in

5063-402: The denominator is a reminder that | ψ ⟩ {\displaystyle \vert \psi \rangle } and likewise | φ ⟩ {\displaystyle \vert \varphi \rangle } were not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be interpreted as the angle between two vectors; thus it

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5146-480: The distance (length of a geodesic) between them is or, equivalently, in projective variety notation, Here, Z ¯ α {\displaystyle {\bar {Z}}^{\alpha }} is the complex conjugate of Z α {\displaystyle Z_{\alpha }} . The appearance of ⟨ ψ | ψ ⟩ {\displaystyle \langle \psi \vert \psi \rangle } in

5229-530: The division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions. A σ -sesquilinear form over a right K -module M is a bi-additive map φ  : M × M → K with an associated anti-automorphism σ of a division ring K such that, for all x , y in M and all α , β in K , The associated anti-automorphism σ for any nonzero sesquilinear form φ

5312-452: The fiber E p {\displaystyle E_{p}} and h p ( ζ , ζ ¯ ) > 0 {\displaystyle h_{p}{\mathord {\left(\zeta ,{\bar {\zeta }}\right)}}>0} for all nonzero ζ {\displaystyle \zeta } in E p {\displaystyle E_{p}} . A Hermitian manifold

5395-412: The following Kähler potential : as An expression is also possible in the notation of homogeneous coordinates , commonly used to describe projective varieties of algebraic geometry : Z  = [ Z 0 :...: Z n ]. Formally, subject to suitably interpreting the expressions involved, one has Here the summation convention is used to sum over Greek indices α β ranging from 0 to n , and in

5478-510: The homogeneous coordinates given above, let where { | e k ⟩ } {\displaystyle \{\vert e_{k}\rangle \}} is a set of orthonormal basis vectors for Hilbert space , the Z k {\displaystyle Z_{k}} are complex numbers, and Z α = [ Z 0 : Z 1 : … : Z n ] {\displaystyle Z_{\alpha }=[Z_{0}:Z_{1}:\ldots :Z_{n}]}

5561-834: The identity h = g − i ω . {\displaystyle h=g-i\omega .} All three forms h , g , and ω preserve the almost complex structure J . That is, h ( J u , J v ) = h ( u , v ) g ( J u , J v ) = g ( u , v ) ω ( J u , J v ) = ω ( u , v ) {\displaystyle {\begin{aligned}h(Ju,Jv)&=h(u,v)\\g(Ju,Jv)&=g(u,v)\\\omega (Ju,Jv)&=\omega (u,v)\end{aligned}}} for all complex tangent vectors u and v . A Hermitian structure on an (almost) complex manifold M can therefore be specified by either Note that many authors call g itself

5644-407: The last equality the standard notation for the skew part of a tensor is used: Now, this expression for d s apparently defines a tensor on the total space of the tautological bundle C \{0}. It is to be understood properly as a tensor on CP by pulling it back along a holomorphic section σ of the tautological bundle of CP . It remains then to verify that the value of the pullback is independent of

5727-432: The manifold. By dropping this condition, we get an almost Hermitian manifold . On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure ) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it is closed (i.e., it is a symplectic form ), we get an almost Kähler structure . If both

5810-437: The map w ↦ φ ( z , w ) {\displaystyle w\mapsto \varphi (z,w)} is a linear functional on V {\displaystyle V} (i.e. an element of the dual space V ∗ {\displaystyle V^{*}} ). Likewise, the map w ↦ φ ( w , z ) {\displaystyle w\mapsto \varphi (w,z)}

5893-526: The more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear. Over a complex vector space V {\displaystyle V} a map φ : V × V → C {\displaystyle \varphi :V\times V\to \mathbb {C} } is sesquilinear if for all x , y , z , w ∈ V {\displaystyle x,y,z,w\in V} and all

5976-464: The nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the arbitrary field version of the definition to arbitrary rings. Let R be a ring , V an R - module and σ an antiautomorphism of R . A map φ  : V × V → R is σ -sesquilinear if for all x , y , z , w in V and all c , d in R . An element x

6059-439: The obvious manner. The n +1 coordinate patches U i {\displaystyle U_{i}} cover CP , and it is possible to give the metric explicitly in terms of the affine coordinates ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} on U i {\displaystyle U_{i}} . The coordinate derivatives define

6142-495: The quotient in (a) is the real hypersphere S defined by the equation | Z | = | Z 0 | + ... + | Z n | = 1. The quotient in (b) realizes CP  =  S / S , where S represents the group of rotations. This quotient is realized explicitly by the famous Hopf fibration S  →  S  →  CP , the fibers of which are among the great circles of S 2 n + 1 {\displaystyle S^{2n+1}} . When

6225-401: The quotient of C \{0} by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group C  =  C  \ {0}: This quotient realizes C \{0} as a complex line bundle over the base space CP . (In fact this is the so-called tautological bundle over CP .) A point of CP

6308-400: The quotient. However, this metric is invariant under the diagonal action of S  = U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished. The Fubini–Study metric is the metric induced on the quotient CP  =  S / S , where S 2 n + 1 {\displaystyle S^{2n+1}} carries

6391-618: The same then φ {\displaystyle \varphi } is said to be Hermitian . If they are negatives of one another, then φ {\displaystyle \varphi } is said to be skew-Hermitian . Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form. If V {\displaystyle V} is a finite-dimensional complex vector space, then relative to any basis { e i } i {\displaystyle \left\{e_{i}\right\}_{i}} of V , {\displaystyle V,}

6474-446: The scalars to come from any field and the twist is provided by a field automorphism . An application in projective geometry requires that the scalars come from a division ring (skew field), K , and this means that the "vectors" should be replaced by elements of a K -module . In a very general setting, sesquilinear forms can be defined over R -modules for arbitrary rings R . Sesquilinear forms abstract and generalize

6557-469: The second to be linear. This is the convention used mostly by physicists and originates in Dirac's bra–ket notation in quantum mechanics . It is also consistent with the definition of the usual (Euclidean) product of w , z ∈ C n {\displaystyle w,z\in \mathbb {C} ^{n}} as w ∗ z {\displaystyle w^{*}z} . In

6640-446: The sense that for any element h  ∈  G and pair of vector fields X , Y {\displaystyle X,Y} we must have g ( Xh , Yh ) =  g ( X , Y ). The standard Hermitian metric on C is given in the standard basis by whose realification is the standard Euclidean metric on R . This metric is not invariant under the diagonal action of C , so we are unable to directly push it down to CP in

6723-957: The so-called "round metric" endowed upon it by restriction of the standard Euclidean metric to the unit hypersphere. Corresponding to a point in CP with homogeneous coordinates [ Z 0 : ⋯ : Z n ] {\displaystyle [Z_{0}:\dots :Z_{n}]} , there is a unique set of n coordinates ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} such that provided Z 0 ≠ 0 {\displaystyle Z_{0}\neq 0} ; specifically, z j = Z j / Z 0 {\displaystyle z_{j}=Z_{j}/Z_{0}} . The ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} form an affine coordinate system for CP in

6806-419: The usual abbreviations that d r 2 = d r ⊗ d r {\displaystyle dr^{\,2}=dr\otimes dr} and σ k 2 = σ k ⊗ σ k {\displaystyle \sigma _{k}^{\,2}=\sigma _{k}\otimes \sigma _{k}} . The line element, starting with the previously given expression,

6889-802: The vector bundle ( E ⊗ E ¯ ) ∗ {\displaystyle (E\otimes {\overline {E}})^{*}} such that for every point p {\displaystyle p} in M {\displaystyle M} , h p ( η , ζ ¯ ) = h p ( ζ , η ¯ ) ¯ {\displaystyle h_{p}{\mathord {\left(\eta ,{\bar {\zeta }}\right)}}={\overline {h_{p}{\mathord {\left(\zeta ,{\bar {\eta }}\right)}}}}} for all ζ {\displaystyle \zeta } , η {\displaystyle \eta } in

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