Fubini–Study metric

Fubini–Study metric
Name	Fubini–Study metric
Field	Differential geometry, Complex geometry, Quantum mechanics
Introduced	1904
Introduced by	Guido Fubini, Eduard Study

Fubini–Study metric The Fubini–Study metric is a canonical Riemannian metric on complex projective space that encodes a natural distance and curvature structure. It arises from Hermitian forms and symplectic geometry and plays a central role in complex differential geometry, algebraic geometry, and the geometric formulation of quantum mechanics. The metric connects constructions by Guido Fubini and Eduard Study to broader developments in Élie Cartan's theory, Hermann Weyl's work on symmetry, and later applications in John von Neumann's formalism.

Definition and construction

The metric is defined on complex projective space CP^n via a Hermitian inner product on C^{n+1} together with projectivization, following approaches used by Henri Poincaré, Elie Cartan, Kähler, and Shing-Tung Yau. Starting from a nondegenerate Hermitian form invariant under U(n+1), the construction projects the flat metric on C^{n+1} onto CP^n using the quotient by the action of C^* or U(1), as in methods developed by Hermann Weyl and Élie Cartan. Equivalent formulations use the Kähler potential log||z||^2 introduced in studies related to André Weil and Kunihiko Kodaira.

Properties and geometry

The metric is Kähler and Einstein with positive Ricci curvature, properties investigated in work by Shing-Tung Yau, S.-T. Yau, and Eugenio Calabi. Its holomorphic sectional curvature is constant, linking to classification results by Wolfgang Ballmann and Bertram Kostant for symmetric spaces. CP^n endowed with the metric is a compact, simply connected, homogeneous space under PU(n+1), reflecting symmetry principles analyzed by Hermann Weyl and Élie Cartan. Geodesics correspond to great circles in the induced embedding related to Veronese map studies by Oscar Zariski and David Mumford. The metric yields canonical notions of volume and diameter used in comparisons by Marcel Berger and in rigidity theorems by Gromov and Mikhail Gromov.

Expression in homogeneous and inhomogeneous coordinates

In homogeneous coordinates [z_0:...:z_n], the metric derives from the Fubini–Study form ω_FS = i∂∂̄ log∑|z_j|^2, an expression rooted in techniques by Hodge and Weyl. In affine inhomogeneous charts z_k ≠ 0 the local potential is log(1+∑|w_i|^2) mirroring constructions by Kunihiko Kodaira and André Weil. Coordinate expressions facilitate computations in algebraic geometry contexts explored by Alexander Grothendieck, Jean-Pierre Serre, and David Mumford, and match with curvature formulas used in studies by Simon Donaldson and Karen Uhlenbeck.

Relation to projective space and complex geometry

As the standard metric on complex projective space, it exemplifies compact Hermitian symmetric spaces classified by Élie Cartan and links to the theory of ample line bundles via the Chern form of O(1), an idea central to Alexander Grothendieck's and Jean-Pierre Serre's work. It plays a role in embedding theorems by Tian, Yau, and Kodaira and in stability criteria linked to the Mumford stability framework developed by David Mumford and Simon Donaldson. The metric interplays with Hodge theory as in contributions by Phillip Griffiths and Wilfried Schmid and informs moduli problems treated by Pierre Deligne and David Mumford.

Applications in quantum mechanics and information theory

The metric provides a natural distance on the projective Hilbert space of pure states, connecting to the geometric formulations advocated by Paul Dirac, John von Neumann, and Hermann Weyl. Transition probabilities and Fubini–Study distances underlie analyses in quantum information by Asher Peres, Carl Schumacher, Charles Bennett, and Peter Shor, and feature in quantum state discrimination problems studied by Alexander Holevo and Benjamin Schumacher. It appears in quantization schemes such as geometric quantization from work by Bertram Kostant and Jean-Marie Souriau, and in semiclassical approximations used by Mikhail Shubin and Victor Guillemin. In quantum computing, it is used in gate fidelity measures considered by John Preskill and Michael Nielsen.

Generalizations and related metrics

Generalizations include metrics on Grassmannians and flag varieties studied by Bertram Kostant, Robert Bott, and Raoul Bott, and the Bergman metric connected to Siegfried Bergman's kernel. Extensions to infinite-dimensional projective Hilbert spaces connect to work by Israel Gelfand, Mark Krein, and John von Neumann. Further relatives are the Kobayashi metric examined by Shoshichi Kobayashi, the Carathéodory metric considered by László Carathéodory, and balanced metrics studied by Simon Donaldson and Tian. Developments in mirror symmetry and string theory link to contributions by Cumrun Vafa, Edward Witten, and Maxim Kontsevich where related Kähler metrics play roles in moduli stabilization.

Category:Complex differential geometry