Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Fubini–Study metric
open-in-new
<h2 id="construction">Construction</h2> <p>The Fubini–Study metric arises naturally in the <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient space</a> construction of <a href="/facts/Complex_projective_space/QmgneIzS">complex projective space</a>. </p><p>Specifically, one may define CP<i>n</i> to be the space consisting of all complex lines in C<i>n</i>+1, i.e., the quotient of C<i>n</i>+1\{0} by the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> relating all complex multiples of each point together. This agrees with the quotient by the diagonal <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> of the multiplicative group C* = C \ {0}: </p> C P n = { Z = [ Z 0 , Z 1 , … , Z n ] ∈ C n + 1 ∖ { 0 } } / { Z ∼ c Z , c ∈ C ∗ } . {\displaystyle \mathbf {CP} ^{n}=\left\{\mathbf {Z} =[Z_{0},Z_{1},\ldots ,Z_{n}]\in {\mathbf {C} }^{n+1}\setminus \{0\}\right\}{\big /}\{\mathbf {Z} \sim c\mathbf {Z} ,c\in \mathbf {C} ^{*}\}.} <p>This quotient realizes C<i>n</i>+1\{0} as a complex <a href="/facts/Line_bundle/CupAl1ft">line bundle</a> over the base space CP<i>n</i>. (In fact this is the so-called <a href="/facts/Tautological_bundle/Vc2CmshX">tautological bundle</a> over CP<i>n</i>.) A point of CP<i>n</i> is thus identified with an equivalence class of (<i>n</i>+1)-tuples [<i>Z</i>0,...,<i>Z</i><i>n</i>] modulo nonzero complex rescaling; the <i>Z</i><i>i</i> are called <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a> of the point. </p><p>Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar <i>z</i> = <i>R</i> <i>e</i>iθ can be uniquely thought of as the composition of a dilation by the modulus <i>R</i> followed by a counterclockwise rotation about the origin by an angle θ {\displaystyle \theta } , the quotient mapping C<i>n</i>+1 → CP<i>n</i> splits into two pieces. </p> C n + 1 ∖ { 0 } ⟶ ( a ) S 2 n + 1 ⟶ ( b ) C P n {\displaystyle \mathbf {C} ^{n+1}\setminus \{0\}\mathrel {\stackrel {(a)}{\longrightarrow }} S^{2n+1}\mathrel {\stackrel {(b)}{\longrightarrow }} \mathbf {CP} ^{n}} <p>where step (a) is a quotient by the dilation Z ~ <i>R</i>Z for <i>R</i> ∈ R+, the multiplicative group of <a href="/facts/Positive_real_numbers/VFXU56H9">positive real numbers</a>, and step (b) is a quotient by the rotations Z ~ <i>e</i>iθZ. </p><p>The result of the quotient in (a) is the real hypersphere <i>S</i>2<i>n</i>+1 defined by the equation |Z|2 = |<i>Z</i>0|2 + ... + |<i>Z</i><i>n</i>|2 = 1. The quotient in (b) realizes CP<i>n</i> = <i>S</i>2<i>n</i>+1/<i>S</i>1, where <i>S</i>1 represents the group of rotations. This quotient is realized explicitly by the famous <a href="/facts/Hopf_fibration/wRKjQMz0">Hopf fibration</a> <i>S</i>1 → <i>S</i>2<i>n</i>+1 → CP<i>n</i>, the fibers of which are among the <a href="/facts/Great_circles/x4fq6eNm">great circles</a> of S 2 n + 1 {\displaystyle S^{2n+1}} . </p> <h3>As a metric quotient</h3> <p>When a quotient is taken of a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> (or <a href="/facts/Metric_space/gnBBRXtc">metric space</a> in general), care must be taken to ensure that the quotient space is endowed with a <a href="/facts/Riemannian_metric/5KYnmEgF">metric</a> that is well-defined. For instance, if a group <i>G</i> acts on a Riemannian manifold (<i>X</i>,<i>g</i>), then in order for the <a href="/facts/Orbit_space/Vh9VtUUw">orbit space</a> <i>X</i>/<i>G</i> to possess an induced metric, g {\displaystyle g} must be constant along <i>G</i>-orbits in the sense that for any element <i>h</i> ∈ <i>G</i> and pair of vector fields X , Y {\displaystyle X,Y} we must have <i>g</i>(<i>Xh</i>,<i>Yh</i>) = <i>g</i>(<i>X</i>,<i>Y</i>). </p><p>The standard <a href="/facts/Hermitian_metric/69fYok0x">Hermitian metric</a> on C<i>n</i>+1 is given in the standard basis by </p> d s 2 = d Z ⊗ d Z ¯ = d Z 0 ⊗ d Z ¯ 0 + ⋯ + d Z n ⊗ d Z ¯ n {\displaystyle ds^{2}=d\mathbf {Z} \otimes d{\bar {\mathbf {Z} }}=dZ_{0}\otimes d{\bar {Z}}_{0}+\cdots +dZ_{n}\otimes d{\bar {Z}}_{n}} <p>whose realification is the standard <a href="/facts/Euclidean_metric/9qDoQKQe">Euclidean metric</a> on R2<i>n</i>+2. This metric is <i>not</i> invariant under the diagonal action of C*, so we are unable to directly push it down to CPn in the quotient. However, this metric <i>is</i> invariant under the diagonal action of <i>S</i>1 = U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished. </p><p>The Fubini–Study metric is the metric induced on the quotient CP<i>n</i> = <i>S</i>2<i>n</i>+1/<i>S</i>1, where S 2 n + 1 {\displaystyle S^{2n+1}} carries the so-called "round metric" endowed upon it by <i>restriction</i> of the standard Euclidean metric to the unit hypersphere. </p> <h3>In local affine coordinates</h3> <p>Corresponding to a point in CP<i>n</i> with homogeneous coordinates [ Z 0 : ⋯ : Z n ] {\displaystyle [Z_{0}:\dots :Z_{n}]} , there is a unique set of <i>n</i> coordinates ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} such that </p> [ Z 0 : ⋯ : Z n ] ∼ [ 1 , z 1 , … , z n ] , {\displaystyle [Z_{0}:\dots :Z_{n}]\sim [1,z_{1},\dots ,z_{n}],} <p>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 <a href="/facts/Affine_coordinates/tlvI3oX1">affine coordinate system</a> for CP<i>n</i> in 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 the obvious manner. The <i>n</i>+1 coordinate patches U i {\displaystyle U_{i}} cover CP<i>n</i>, 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 a frame { ∂ 1 , … , ∂ n } {\displaystyle \{\partial _{1},\ldots ,\partial _{n}\}} of the holomorphic tangent bundle of CP<i>n</i>, in terms of which the Fubini–Study metric has Hermitian components </p> g i j ¯ = h ( ∂ i , ∂ ¯ j ) = ( 1 + | z | l 2 ) δ i j ¯ − z ¯ i z j ( 1 + | z | l 2 ) 2 . {\displaystyle g_{i{\bar {j}}}=h(\partial _{i},{\bar {\partial }}_{j})={\frac {\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)\delta _{i{\bar {j}}}-{\bar {z}}_{i}z_{j}}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}.} <p>where |z|2 = |<i>z</i>1|2 + ... + |<i>z</i><i>n</i>|2. That is, the <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> of the Fubini–Study metric in this frame is </p> [ g i j ¯ ] = 1 ( 1 + | z | l 2 ) 2 [ 1 + | z | 2 − | z 1 | 2 − z ¯ 1 z 2 ⋯ − z ¯ 1 z n − z ¯ 2 z 1 1 + | z | 2 − | z 2 | 2 ⋯ − z ¯ 2 z n ⋮ ⋮ ⋱ ⋮ − z ¯ n z 1 − z ¯ n z 2 ⋯ 1 + | z | 2 − | z n | 2 ] {\displaystyle {\bigl [}g_{i{\bar {j}}}{\bigr ]}={\frac {1}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}\left[{\begin{array}{cccc}1+|\mathbf {z} |^{2}-|z_{1}|^{2}&-{\bar {z}}_{1}z_{2}&\cdots &-{\bar {z}}_{1}z_{n}\\-{\bar {z}}_{2}z_{1}&1+|\mathbf {z} |^{2}-|z_{2}|^{2}&\cdots &-{\bar {z}}_{2}z_{n}\\\vdots &\vdots &\ddots &\vdots \\-{\bar {z}}_{n}z_{1}&-{\bar {z}}_{n}z_{2}&\cdots &1+|\mathbf {z} |^{2}-|z_{n}|^{2}\end{array}}\right]} <p>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. </p><p>Accordingly, the line element is given by </p> d s 2 = g i j ¯ d z i d z ¯ j = ( 1 + | z | l 2 ) | d z | 2 − ( z ¯ ⋅ d z ) ( z ⋅ d z ¯ ) ( 1 + | z | l 2 ) 2 = ( 1 + z i z ¯ i ) d z j d z ¯ j − z ¯ j z i d z j d z ¯ i ( 1 + z i z ¯ i ) 2 . {\displaystyle {\begin{aligned}ds^{2}&=g_{i{\bar {j}}}\,dz^{i}\,d{\bar {z}}^{j}\\[4pt]&={\frac {\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)|d\mathbf {z} |^{2}-({\bar {\mathbf {z} }}\cdot d\mathbf {z} )(\mathbf {z} \cdot d{\bar {\mathbf {z} }})}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}\\[4pt]&={\frac {(1+z_{i}{\bar {z}}^{i})\,dz_{j}\,d{\bar {z}}^{j}-{\bar {z}}^{j}z_{i}\,dz_{j}\,d{\bar {z}}^{i}}{\left(1+z_{i}{\bar {z}}^{i}\right)^{2}}}.\end{aligned}}} <p>In this last expression, the <a href="/facts/Summation_convention/UXwS43wU">summation convention</a> is used to sum over Latin indices <i>i</i>,<i>j</i> that range from 1 to <i>n</i>. </p><p>The metric can be derived from the following <a href="/facts/K%25C3%25A4hler_potential/JFPM6Vxg">Kähler potential</a>:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> K = ln ( 1 + z i z ¯ i ) = ln ( 1 + δ i j ¯ z i z ¯ j ) {\displaystyle K=\ln(1+z_{i}{\bar {z}}^{i})=\ln(1+\delta _{i{\bar {j}}}z^{i}{\bar {z}}^{j})} <p>as </p> g i j ¯ = K i j ¯ = ∂ 2 ∂ z i ∂ z ¯ j K {\displaystyle g_{i{\bar {j}}}=K_{i{\bar {j}}}={\frac {\partial ^{2}}{\partial z^{i}\,\partial {\bar {z}}^{j}}}K} <h3>Using homogeneous coordinates</h3> <p>An expression is also possible in the notation of <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a>, commonly used to describe <a href="/facts/Projective_varieties/aB2XdB7B">projective varieties</a> of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>: Z = [<i>Z</i>0:...:<i>Z</i><i>n</i>]. Formally, subject to suitably interpreting the expressions involved, one has </p> d s 2 = | Z | 2 | d Z | 2 − ( Z ¯ ⋅ d Z ) ( Z ⋅ d Z ¯ ) | Z | 4 = Z α Z ¯ α d Z β d Z ¯ β − Z ¯ α Z β d Z α d Z ¯ β ( Z α Z ¯ α ) 2 = 2 Z [ α d Z β ] Z ¯ [ α d Z ¯ β ] ( Z α Z ¯ α ) 2 . {\displaystyle {\begin{aligned}ds^{2}&={\frac {|\mathbf {Z} |^{2}|d\mathbf {Z} |^{2}-({\bar {\mathbf {Z} }}\cdot d\mathbf {Z} )(\mathbf {Z} \cdot d{\bar {\mathbf {Z} }})}{|\mathbf {Z} |^{4}}}\\&={\frac {Z_{\alpha }{\bar {Z}}^{\alpha }dZ_{\beta }d{\bar {Z}}^{\beta }-{\bar {Z}}^{\alpha }Z_{\beta }dZ_{\alpha }d{\bar {Z}}^{\beta }}{\left(Z_{\alpha }{\bar {Z}}^{\alpha }\right)^{2}}}\\&={\frac {2Z_{[\alpha }\,dZ_{\beta ]}{\bar {Z}}^{[\alpha }\,{\overline {dZ}}^{\beta ]}}{\left(Z_{\alpha }{\bar {Z}}^{\alpha }\right)^{2}}}.\end{aligned}}} <p>Here the summation convention is used to sum over Greek indices α β ranging from 0 to <i>n</i>, and in the last equality the standard notation for the skew part of a tensor is used: </p> Z [ α W β ] = 1 2 ( Z α W β − Z β W α ) . {\displaystyle Z_{[\alpha }W_{\beta ]}={\tfrac {1}{2}}\left(Z_{\alpha }W_{\beta }-Z_{\beta }W_{\alpha }\right).} <p>Now, this expression for d<i>s</i>2 apparently defines a tensor on the total space of the tautological bundle C<i>n</i>+1\{0}. It is to be understood properly as a tensor on CP<i>n</i> by pulling it back along a holomorphic section σ of the tautological bundle of CP<i>n</i>. It remains then to verify that the value of the pullback is independent of the choice of section: this can be done by a direct calculation. </p><p>The <a href="/facts/K%25C3%25A4hler_form/JFPM6Vxg">Kähler form</a> of this metric is </p> ω = i 2 ∂ ∂ ¯ log | Z | 2 {\displaystyle \omega ={\frac {i}{2}}\partial {\bar {\partial }}\log |\mathbf {Z} |^{2}} <p>where the ∂ , ∂ ¯ {\displaystyle \partial ,{\bar {\partial }}} are the <a href="/facts/Dolbeault_operators/307vM8pd">Dolbeault operators</a>. The pullback of this is clearly independent of the choice of holomorphic section. The quantity log|Z|2 is the <a href="/facts/K%25C3%25A4hler_potential/JFPM6Vxg">Kähler potential</a> (sometimes called the Kähler scalar) of CP<i>n</i>. </p> <h3>In bra-ket coordinate notation</h3> <p>In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the Fubini–Study metric is also known as the <a href="/facts/Bures_metric/BCDbPPPJ">Bures metric</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> However, the Bures metric is typically defined in the notation of <a href="/facts/Mixed_state_(physics)/7jbxd7Dx">mixed states</a>, whereas the exposition below is written in terms of a <a href="/facts/Pure_state/7jbxd7Dx">pure state</a>. The real part of the metric is (a quarter of) the <a href="/facts/Fisher_information_metric/zEeAWdII">Fisher information metric</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>The Fubini–Study metric may be written using the <a href="/facts/Bra%25E2%2580%2593ket_notation/NDlKsaRh">bra–ket notation</a> commonly used in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. To explicitly equate this notation to the homogeneous coordinates given above, let </p> | ψ ⟩ = ∑ k = 0 n Z k | e k ⟩ = [ Z 0 : Z 1 : … : Z n ] {\displaystyle \vert \psi \rangle =\sum _{k=0}^{n}Z_{k}\vert e_{k}\rangle =[Z_{0}:Z_{1}:\ldots :Z_{n}]} <p>where { | e k ⟩ } {\displaystyle \{\vert e_{k}\rangle \}} is a set of <a href="/facts/Orthonormal/JfgL1nAh">orthonormal</a> <a href="/facts/Basis_vector/89IPoN6c">basis vectors</a> for <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, 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}]} is the standard notation for a point in the <a href="/facts/Projective_space/7MSpA9cM">projective space</a> CP<i>n</i> in <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a>. 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, the distance (length of a geodesic) between them is </p> γ ( ψ , φ ) = arccos ⟨ ψ | φ ⟩ ⟨ φ | ψ ⟩ ⟨ ψ | ψ ⟩ ⟨ φ | φ ⟩ {\displaystyle \gamma (\psi ,\varphi )=\arccos {\sqrt {\frac {\langle \psi \vert \varphi \rangle \;\langle \varphi \vert \psi \rangle }{\langle \psi \vert \psi \rangle \;\langle \varphi \vert \varphi \rangle }}}} <p>or, equivalently, in projective variety notation, </p> γ ( ψ , φ ) = γ ( Z , W ) = arccos Z α W ¯ α W β Z ¯ β Z α Z ¯ α W β W ¯ β . {\displaystyle \gamma (\psi ,\varphi )=\gamma (Z,W)=\arccos {\sqrt {\frac {Z_{\alpha }{\bar {W}}^{\alpha }\;W_{\beta }{\bar {Z}}^{\beta }}{Z_{\alpha }{\bar {Z}}^{\alpha }\;W_{\beta }{\bar {W}}^{\beta }}}}.} <p>Here, Z ¯ α {\displaystyle {\bar {Z}}^{\alpha }} is the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of Z α {\displaystyle Z_{\alpha }} . The appearance of ⟨ ψ | ψ ⟩ {\displaystyle \langle \psi \vert \psi \rangle } in 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 is occasionally called the quantum angle. The angle is real-valued, and runs from 0 to π / 2 {\displaystyle \pi /2} . </p><p>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 </p> d s 2 = ⟨ δ ψ | δ ψ ⟩ ⟨ ψ | ψ ⟩ − ⟨ δ ψ | ψ ⟩ ⟨ ψ | δ ψ ⟩ ⟨ ψ | ψ ⟩ 2 . {\displaystyle ds^{2}={\frac {\langle \delta \psi \vert \delta \psi \rangle }{\langle \psi \vert \psi \rangle }}-{\frac {\langle \delta \psi \vert \psi \rangle \;\langle \psi \vert \delta \psi \rangle }{{\langle \psi \vert \psi \rangle }^{2}}}.} <p>In the context of <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, CP1 is called the <a href="/facts/Bloch_sphere/CvgaFMNf">Bloch sphere</a>; the Fubini–Study metric is the natural <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, including <a href="/facts/Quantum_entanglement/afzslwN0">quantum entanglement</a> and the <a href="/facts/Berry_phase/d0QbNVdS">Berry phase</a> effect, can be attributed to the peculiarities of the Fubini–Study metric. </p> <h2 id="the-n-1-case">The <i>n</i> = 1 case</h2> <p>When <i>n</i> = 1, there is a diffeomorphism S 2 ≅ C P 1 {\displaystyle S^{2}\cong \mathbf {CP} ^{1}} given by <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a>. This leads to the "special" Hopf fibration <i>S</i>1 → <i>S</i>3 → <i>S</i>2. When the Fubini–Study metric is written in coordinates on CP1, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (and <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> 4) on <i>S</i>2. </p><p>Namely, if <i>z</i> = <i>x</i> + i<i>y</i> is the standard affine coordinate chart on the <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a> CP1 and <i>x</i> = <i>r</i> cos θ, <i>y</i> = <i>r</i> sin θ are polar coordinates on C, then a routine computation shows </p> d s 2 = Re ( d z ⊗ d z ¯ ) ( 1 + | z | l 2 ) 2 = d x 2 + d y 2 ( 1 + r 2 ) 2 = 1 4 ( d φ 2 + sin 2 φ d θ 2 ) = 1 4 d s u s 2 {\displaystyle ds^{2}={\frac {\operatorname {Re} (dz\otimes d{\bar {z}})}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}={\frac {dx^{2}+dy^{2}}{\left(1+r^{2}\right)^{2}}}={\tfrac {1}{4}}(d\varphi ^{2}+\sin ^{2}\varphi \,d\theta ^{2})={\tfrac {1}{4}}\,ds_{us}^{2}} <p>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 <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>" on <i>S</i>2 coming from the stereographic projection <i>r</i> tan(φ/2) = 1, tan θ = <i>y</i>/<i>x</i>. (Many physics references interchange the roles of φ and θ.) </p><p>The <a href="/facts/K%25C3%25A4hler_form/JFPM6Vxg">Kähler form</a> is </p> K = i 2 d z ∧ d z ¯ ( 1 + z z ¯ ) 2 = d x ∧ d y ( 1 + x 2 + y 2 ) 2 {\displaystyle K={\frac {i}{2}}{\frac {dz\wedge d{\bar {z}}}{\left(1+z{\bar {z}}\right)^{2}}}={\frac {dx\wedge dy}{\left(1+x^{2}+y^{2}\right)^{2}}}} <p>Choosing as <a href="/facts/Vierbein/jyEhSaHM">vierbeins</a> 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 </p> K = e 1 ∧ e 2 {\displaystyle K=e^{1}\wedge e^{2}} <p>Applying the <a href="/facts/Hodge_star/AH6tmHhU">Hodge star</a> to the Kähler form, one obtains </p> ∗ K = 1 {\displaystyle *K=1} <p>implying that <i>K</i> is <a href="/facts/Harmonic_form/b6Mju74J">harmonic</a>. </p> <h2 id="the-n-2-case">The <i>n</i> = 2 case</h2> <p>The Fubini–Study metric on the <a href="/facts/Complex_projective_plane/5WFpDyeV">complex projective plane</a> CP2 has been proposed as a <a href="/facts/Gravitational_instanton/82C1Xhto">gravitational instanton</a>, the gravitational analog of an <a href="/facts/Instanton/IuXhVcRF">instanton</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> 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 the <a href="/facts/N-sphere/bFu2S7E2">4-sphere</a> (the <a href="/facts/Quaternionic_projective_line/HbQDIvj8">quaternionic projective line</a>) as </p> r d r = + x d x + y d y + z d z + t d t r 2 σ 1 = − t d x − z d y + y d z + x d t r 2 σ 2 = + z d x − t d y − x d z + y d t r 2 σ 3 = − y d x + x d y − t d z + z d t {\displaystyle {\begin{aligned}r\,dr&=+x\,dx+y\,dy+z\,dz+t\,dt\\r^{2}\sigma _{1}&=-t\,dx-z\,dy+y\,dz+x\,dt\\r^{2}\sigma _{2}&=+z\,dx-t\,dy-x\,dz+y\,dt\\r^{2}\sigma _{3}&=-y\,dx+x\,dy-t\,dz+z\,dt\end{aligned}}} <p>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. </p><p>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 </p> z 1 z ¯ 1 + z 2 z ¯ 2 = r 2 = x 2 + y 2 + z 2 + t 2 d z 1 d z ¯ 1 + d z 2 d z ¯ 2 = d r 2 + r 2 ( σ 1 2 + σ 2 2 + σ 3 2 ) z ¯ 1 d z 1 + z ¯ 2 d z 2 = r d r + i r 2 σ 3 {\displaystyle {\begin{aligned}z_{1}{\bar {z}}_{1}+z_{2}{\bar {z}}_{2}&=r^{2}=x^{2}+y^{2}+z^{2}+t^{2}\\dz_{1}\,d{\bar {z}}_{1}+dz_{2}\,d{\bar {z}}_{2}&=dr^{\,2}+r^{2}(\sigma _{1}^{\,2}+\sigma _{2}^{\,2}+\sigma _{3}^{\,2})\\{\bar {z}}_{1}\,dz_{1}+{\bar {z}}_{2}\,dz_{2}&=rdr+i\,r^{2}\sigma _{3}\end{aligned}}} <p>with 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}} . </p><p>The line element, starting with the previously given expression, is given by </p> d s 2 = d z j d z ¯ j 1 + z i z ¯ i − z ¯ j z i d z j d z ¯ i ( 1 + z i z ¯ i ) 2 = d r 2 + r 2 ( σ 1 2 + σ 2 2 + σ 3 2 ) 1 + r 2 − r 2 d r 2 + r 4 σ 3 2 ( 1 + r 2 ) 2 = d r 2 + r 2 σ 3 2 ( 1 + r 2 ) 2 + r 2 ( σ 1 2 + σ 2 2 ) 1 + r 2 {\displaystyle {\begin{aligned}ds^{2}&={\frac {dz_{j}\,d{\bar {z}}^{j}}{1+z_{i}{\bar {z}}^{i}}}-{\frac {{\bar {z}}^{j}z_{i}\,dz_{j}\,d{\bar {z}}^{i}}{(1+z_{i}{\bar {z}}^{i})^{2}}}\\[5pt]&={\frac {dr^{\,2}+r^{2}(\sigma _{1}^{\,2}+\sigma _{2}^{\,2}+\sigma _{3}^{\,2})}{1+r^{2}}}-{\frac {r^{2}dr^{\,2}+r^{4}\sigma _{3}^{\,2}}{\left(1+r^{2}\right)^{2}}}\\[4pt]&={\frac {dr^{\,2}+r^{2}\sigma _{3}^{\,2}}{\left(1+r^{2}\right)^{2}}}+{\frac {r^{2}\left(\sigma _{1}^{\,2}+\sigma _{2}^{\,2}\right)}{1+r^{2}}}\end{aligned}}} <p>The <a href="/facts/Vierbein/jyEhSaHM">vierbeins</a> can be immediately read off from the last expression: </p> e 0 = d r 1 + r 2 e 3 = r σ 3 1 + r 2 e 1 = r σ 1 1 + r 2 e 2 = r σ 2 1 + r 2 {\displaystyle {\begin{aligned}e^{0}={\frac {dr}{1+r^{2}}}&&&e^{3}={\frac {r\sigma _{3}}{1+r^{2}}}\\[5pt]e^{1}={\frac {r\sigma _{1}}{\sqrt {1+r^{2}}}}&&&e^{2}={\frac {r\sigma _{2}}{\sqrt {1+r^{2}}}}\end{aligned}}} <p>That is, in the vierbein coordinate system, using roman-letter subscripts, the metric tensor is Euclidean: </p> d s 2 = δ a b e a ⊗ e b = e 0 ⊗ e 0 + e 1 ⊗ e 1 + e 2 ⊗ e 2 + e 3 ⊗ e 3 . {\displaystyle ds^{2}=\delta _{ab}e^{a}\otimes e^{b}=e^{0}\otimes e^{0}+e^{1}\otimes e^{1}+e^{2}\otimes e^{2}+e^{3}\otimes e^{3}.} <p>Given the vierbein, a <a href="/facts/Spin_connection/GdnYxk3f">spin connection</a> can be computed; the Levi-Civita spin connection is the unique connection that is <a href="/facts/Torsion_form/LVfDhPnW">torsion-free</a> and covariantly constant, namely, it is the one-form ω b a {\displaystyle \omega _{\;\;b}^{a}} that satisfies the torsion-free condition </p> d e a + ω b a ∧ e b = 0 {\displaystyle de^{a}+\omega _{\;\;b}^{a}\wedge e^{b}=0} <p>and is covariantly constant, which, for spin connections, means that it is antisymmetric in the vierbein indexes: </p> ω a b = − ω b a {\displaystyle \omega _{ab}=-\omega _{ba}} <p>The above is readily solved; one obtains </p> ω 1 0 = − ω 3 2 = − e 1 r ω 2 0 = − ω 1 3 = − e 2 r ω 3 0 = r 2 − 1 r e 3 ω 2 1 = 1 + 2 r 2 r e 3 {\displaystyle {\begin{aligned}\omega _{\;\;1}^{0}&=-\omega _{\;\;3}^{2}=-{\frac {e^{1}}{r}}\\\omega _{\;\;2}^{0}&=-\omega _{\;\;1}^{3}=-{\frac {e^{2}}{r}}\\\omega _{\;\;3}^{0}&={\frac {r^{2}-1}{r}}e^{3}\quad \quad \omega _{\;\;2}^{1}={\frac {1+2r^{2}}{r}}e^{3}\\\end{aligned}}} <p>The <a href="/facts/Riemann_curvature_tensor/KyeHCWEh">curvature 2-form</a> is defined as </p> R b a = d ω b a + ω c a ∧ ω b c {\displaystyle R_{\;\,b}^{a}=d\omega _{\;\,b}^{a}+\omega _{\;c}^{a}\wedge \omega _{\;\,b}^{c}} <p>and is constant: </p> R 01 = − R 23 = e 0 ∧ e 1 − e 2 ∧ e 3 R 02 = − R 31 = e 0 ∧ e 2 − e 3 ∧ e 1 R 03 = 4 e 0 ∧ e 3 + 2 e 1 ∧ e 2 R 12 = 2 e 0 ∧ e 3 + 4 e 1 ∧ e 2 {\displaystyle {\begin{aligned}R_{01}&=-R_{23}=e^{0}\wedge e^{1}-e^{2}\wedge e^{3}\\R_{02}&=-R_{31}=e^{0}\wedge e^{2}-e^{3}\wedge e^{1}\\R_{03}&=4e^{0}\wedge e^{3}+2e^{1}\wedge e^{2}\\R_{12}&=2e^{0}\wedge e^{3}+4e^{1}\wedge e^{2}\end{aligned}}} <p>The <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a> in veirbein indexes is given by </p> Ric c a = R b c d a δ b d {\displaystyle \operatorname {Ric} _{\;\;c}^{a}=R_{\;\,bcd}^{a}\delta ^{bd}} <p>where the curvature 2-form was expanded as a four-component tensor: </p> R b a = 1 2 R b c d a e c ∧ e d {\displaystyle R_{\;\,b}^{a}={\tfrac {1}{2}}R_{\;\,bcd}^{a}e^{c}\wedge e^{d}} <p>The resulting <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a> is constant </p> Ric a b = 6 δ a b {\displaystyle \operatorname {Ric} _{ab}=6\delta _{ab}} <p>so that the resulting <a href="/facts/Einstein_equation/j2rJTzW9">Einstein equation</a> </p> Ric a b − 1 2 δ a b R + Λ δ a b = 0 {\displaystyle \operatorname {Ric} _{ab}-{\tfrac {1}{2}}\delta _{ab}R+\Lambda \delta _{ab}=0} <p>can be solved with the <a href="/facts/Cosmological_constant/bjMksSA2">cosmological constant</a> Λ = 6 {\displaystyle \Lambda =6} . </p><p>The <a href="/facts/Weyl_tensor/lrXNLWHC">Weyl tensor</a> for Fubini–Study metrics in general is given by </p> W a b c d = R a b c d − 2 ( δ a c δ b d − δ a d δ b c ) {\displaystyle W_{abcd}=R_{abcd}-2\left(\delta _{ac}\delta _{bd}-\delta _{ad}\delta _{bc}\right)} <p>For the <i>n</i> = 2 case, the two-forms </p> W a b = 1 2 W a b c d e c ∧ e d {\displaystyle W_{ab}={\tfrac {1}{2}}W_{abcd}e^{c}\wedge e^{d}} <p>are self-dual: </p> W 01 = W 23 = − e 0 ∧ e 1 − e 2 ∧ e 3 W 02 = W 31 = − e 0 ∧ e 2 − e 3 ∧ e 1 W 03 = W 12 = 2 e 0 ∧ e 3 + 2 e 1 ∧ e 2 {\displaystyle {\begin{aligned}W_{01}&=W_{23}=-e^{0}\wedge e^{1}-e^{2}\wedge e^{3}\\W_{02}&=W_{31}=-e^{0}\wedge e^{2}-e^{3}\wedge e^{1}\\W_{03}&=W_{12}=2e^{0}\wedge e^{3}+2e^{1}\wedge e^{2}\end{aligned}}} <h2 id="curvature-properties">Curvature properties</h2> <p>In the <i>n</i> = 1 special case, the Fubini–Study metric has constant sectional curvature identically equal to 4, according to the equivalence with the 2-sphere's round metric (which given a radius <i>R</i> has sectional curvature 1 / R 2 {\displaystyle 1/R^{2}} ). However, for <i>n</i> > 1, the Fubini–Study metric does not have constant curvature. Its sectional curvature is instead given by the equation<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> K ( σ ) = 1 + 3 ⟨ J X , Y ⟩ 2 {\displaystyle K(\sigma )=1+3\langle JX,Y\rangle ^{2}} <p>where { X , Y } ∈ T p C P n {\displaystyle \{X,Y\}\in T_{p}\mathbf {CP} ^{n}} is an orthonormal basis of the 2-plane σ, the mapping <i>J</i> : <i>T</i>CP<i>n</i> → <i>T</i>CP<i>n</i> is the <a href="/facts/Linear_complex_structure/gzGklcAX">complex structure</a> on CP<i>n</i>, and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the Fubini–Study metric. </p><p>A consequence of this formula is that the sectional curvature satisfies 1 ≤ K ( σ ) ≤ 4 {\displaystyle 1\leq K(\sigma )\leq 4} for all 2-planes σ {\displaystyle \sigma } . The maximum sectional curvature (4) is attained at a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> 2-plane — one for which <i>J</i>(σ) ⊂ σ — while the minimum sectional curvature (1) is attained at a 2-plane for which <i>J</i>(σ) is orthogonal to σ. For this reason, the Fubini–Study metric is often said to have "constant <i>holomorphic</i> sectional curvature" equal to 4. </p><p>This makes CP<i>n</i> a (non-strict) <a href="/facts/Quarter-pinched_sphere_theorem/sMq3gAxC">quarter pinched manifold</a>; a celebrated theorem shows that a strictly quarter-pinched <a href="/facts/Simply_connected/xFlta63J">simply connected</a> <i>n</i>-manifold must be homeomorphic to a sphere. </p><p>The Fubini–Study metric is also an <a href="/facts/Einstein_metric/HwYVckeu">Einstein metric</a> in that it is proportional to its own <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a>: there exists a constant Λ {\displaystyle \Lambda } ; such that for all <i>i</i>,<i>j</i> we have </p> Ric i j = Λ g i j . {\displaystyle \operatorname {Ric} _{ij}=\Lambda g_{ij}.} <p>This implies, among other things, that the Fubini–Study metric remains unchanged up to a scalar multiple under the <a href="/facts/Ricci_flow/J6rQd3dm">Ricci flow</a>. It also makes CP<i>n</i> indispensable to the theory of <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, where it serves as a nontrivial solution to the vacuum <a href="/facts/Einstein_field_equations/j2rJTzW9">Einstein field equations</a>. </p><p>The <a href="/facts/Cosmological_constant/bjMksSA2">cosmological constant</a> Λ {\displaystyle \Lambda } for CP<i>n</i> is given in terms of the dimension of the space: </p> Ric i j = 2 ( n + 1 ) g i j . {\displaystyle \operatorname {Ric} _{ij}=2(n+1)g_{ij}.} <h2 id="product-metric">Product metric</h2> <p>The common notions of separability apply for the Fubini–Study metric. More precisely, the metric is separable on the natural product of projective spaces, the <a href="/facts/Segre_embedding/xDVUkB2C">Segre embedding</a>. That is, if | ψ ⟩ {\displaystyle \vert \psi \rangle } is a <a href="/facts/Separable_state/j0k8VWmF">separable state</a>, so that it can be written as | ψ ⟩ = | ψ A ⟩ ⊗ | ψ B ⟩ {\displaystyle \vert \psi \rangle =\vert \psi _{A}\rangle \otimes \vert \psi _{B}\rangle } , then the metric is the sum of the metric on the subspaces: </p> d s 2 = d s A 2 + d s B 2 {\displaystyle ds^{2}={ds_{A}}^{2}+{ds_{B}}^{2}} <p>where d s A 2 {\displaystyle {ds_{A}}^{2}} and d s B 2 {\displaystyle {ds_{B}}^{2}} are the metrics, respectively, on the subspaces <i>A</i> and <i>B</i>. </p> <h2 id="connection-and-curvature">Connection and curvature</h2> <p>The fact that the metric can be derived from the Kähler potential means that the <a href="/facts/Christoffel_symbol/uCyd7RWX">Christoffel symbols</a> and the curvature tensors contain a lot of symmetries, and can be given a particularly simple form: The Christoffel symbols, in the local affine coordinates, are given by </p> Γ j k i = g i m ¯ ∂ g k m ¯ ∂ z j Γ j ¯ k ¯ i ¯ = g i ¯ m ∂ g k ¯ m ∂ z ¯ j ¯ {\displaystyle \Gamma _{\;jk}^{i}=g^{i{\bar {m}}}{\frac {\partial g_{k{\bar {m}}}}{\partial z^{j}}}\qquad \Gamma _{\;{\bar {j}}{\bar {k}}}^{\bar {i}}=g^{{\bar {i}}m}{\frac {\partial g_{{\bar {k}}m}}{\partial {\bar {z}}^{\bar {j}}}}} <p>The Riemann tensor is also particularly simple: </p> R i j ¯ k l ¯ = g i m ¯ ∂ Γ j ¯ l ¯ m ¯ ∂ z k {\displaystyle R_{i{\bar {j}}k{\bar {l}}}=g^{i{\bar {m}}}{\frac {\partial \Gamma _{\;\;{\bar {j}}{\bar {l}}}^{\bar {m}}}{\partial z^{k}}}} <p>The <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a> is </p> R i ¯ j = R i ¯ k ¯ j k ¯ = − ∂ Γ i ¯ k ¯ k ¯ ∂ z j R i j ¯ = R i k j ¯ k = − ∂ Γ i k k ∂ z ¯ j ¯ {\displaystyle R_{{\bar {i}}j}=R_{\;{\bar {i}}{\bar {k}}j}^{\bar {k}}=-{\frac {\partial \Gamma _{\;{\bar {i}}{\bar {k}}}^{\bar {k}}}{\partial z^{j}}}\qquad R_{i{\bar {j}}}=R_{\;ik{\bar {j}}}^{k}=-{\frac {\partial \Gamma _{\;ik}^{k}}{\partial {\bar {z}}^{\bar {j}}}}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Non-linear_sigma_model/WqQI5CRJ">Non-linear sigma model</a></li> <li><a href="/facts/Kaluza%25E2%2580%2593Klein_theory/GjfwsA7V">Kaluza–Klein theory</a></li> <li><a href="/facts/Arakelov_height/ByeW10AJ">Arakelov height</a></li></ul> <ul><li><a href="/facts/Arthur_Besse/NDeUz75y">Besse, Arthur L.</a> (1987), <i>Einstein manifolds</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. xii+510, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-15279-8</li> <li>Brody, D.C.; Hughston, L.P. (2001), "Geometric Quantum Mechanics", <i>Journal of Geometry and Physics</i>, 38 (1): 19–53, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/quant-ph/9906086">quant-ph/9906086</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2001JGP....38...19B">2001JGP....38...19B</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0393-0440%2800%2900052-8">10.1016/S0393-0440(00)00052-8</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:17580350">17580350</a></li> <li><a href="/facts/Phillip_Griffiths/EHQJfBVI">Griffiths, P.</a>; <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Harris, J.</a> (1994), <i>Principles of Algebraic Geometry</i>, Wiley Classics Library, Wiley Interscience, pp. 30–31, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-05059-8</li> <li>Onishchik, A.L. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Fubini–Study_metric">"Fubini–Study metric"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, 63 pp. 501–513 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Study, E. (1905). "Kürzeste Wege im komplexen Gebiet". Mathematische Annalen (in German). 60 (3). Springer Science and Business Media LLC: 321–378. doi:10.1007/bf01457616. ISSN 0025-5831. S2CID 120961275. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Eguchi, Tohru; Gilkey, Peter B.; Hanson, Andrew J. (1980). "Gravitation, gauge theories and differential geometry". Physics Reports. 66 (6). Elsevier BV: 213–393. Bibcode:1980PhR....66..213E. doi:10.1016/0370-1573(80)90130-1. ISSN 0370-1573. <a href="https://www.researchgate.net/publication/234195796" target="_blank">https://www.researchgate.net/publication/234195796</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia "Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics" (2010), Physics Letters A 374 pp. 4801. doi:10.1016/j.physleta.2010.10.005 <a href="https://arxiv.org/abs/1009.5219" target="_blank">https://arxiv.org/abs/1009.5219</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia "Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics" (2010), Physics Letters A 374 pp. 4801. doi:10.1016/j.physleta.2010.10.005 <a href="https://arxiv.org/abs/1009.5219" target="_blank">https://arxiv.org/abs/1009.5219</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Eguchi, Tohru; Freund, Peter G. O. (1976-11-08). "Quantum Gravity and World Topology". Physical Review Letters. 37 (19). American Physical Society (APS): 1251–1254. Bibcode:1976PhRvL..37.1251E. doi:10.1103/physrevlett.37.1251. ISSN 0031-9007. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Eguchi, Tohru; Gilkey, Peter B.; Hanson, Andrew J. (1980). "Gravitation, gauge theories and differential geometry". Physics Reports. 66 (6). Elsevier BV: 213–393. Bibcode:1980PhR....66..213E. doi:10.1016/0370-1573(80)90130-1. ISSN 0370-1573. <a href="https://www.researchgate.net/publication/234195796" target="_blank">https://www.researchgate.net/publication/234195796</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Sakai, T. Riemannian Geometry, Translations of Mathematical Monographs No. 149 (1995), American Mathematics Society. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>