Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Complex_projective_space> ?p ?o }
- Complex_projective_space abstract "In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cn+1), Pn(C) or CPn. When n = 1, the complex projective space CP1 is the Riemann sphere, and when n = 2, CP2 is the complex projective plane (see there for a more elementary discussion).Complex projective space was first introduced by von Staudt (1860) as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. Subsequently, near the turn of the 20th century it became clear to the Italian school of algebraic geometry that the complex projective spaces were the most natural domains in which to consider the solutions of polynomial equations - algebraic varieties (Grattan-Guinness 2005, pp. 445–446). In modern times, both the topology and geometry of complex projective space are well-understood and closely related to that of the sphere. Indeed, in a certain sense the (2n+1)-sphere can be regarded as a family of circles parametrized by CPn: this is the Hopf fibration. Complex projective space carries a (Kähler) metric, called the Fubini–Study metric, in terms of which it is a Hermitian symmetric space of rank 1.Complex projective space has many applications in both mathematics and quantum physics. In algebraic geometry, complex projective space is the home of projective varieties, a well-behaved class of algebraic varieties. In topology, the complex projective space plays an important role as a classifying space for complex line bundles: families of complex lines parametrized by another space. In this context, the infinite union of projective spaces (direct limit), denoted CP∞, is the classifying space K(Z,2). In quantum physics, the wave function associated to a pure state of a quantum mechanical system is a probability amplitude, meaning that it has unit norm, and has an inessential overall phase: that is, the wave function of a pure state is naturally a point in the projective Hilbert space of the state space.".
- Complex_projective_space thumbnail Stereographic_projection_in_3D.png?width=300.
- Complex_projective_space wikiPageExternalLink 10.1007%2Fs11511-008-0022-7.
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- Complex_projective_space wikiPageRevisionID "656451946".
- Complex_projective_space wikiPageWikiLink A_fortiori.
- Complex_projective_space wikiPageWikiLink A_fortiori_argument.
- Complex_projective_space wikiPageWikiLink Acta_Mathematica.
- Complex_projective_space wikiPageWikiLink Algebraic_geometry.
- Complex_projective_space wikiPageWikiLink Algebraic_geometry_and_analytic_geometry.
- Complex_projective_space wikiPageWikiLink Algebraic_topology.
- Complex_projective_space wikiPageWikiLink Algebraic_variety.
- Complex_projective_space wikiPageWikiLink BU(1).
- Complex_projective_space wikiPageWikiLink Betti_number.
- Complex_projective_space wikiPageWikiLink Bott_periodicity.
- Complex_projective_space wikiPageWikiLink Bott_periodicity_theorem.
- Complex_projective_space wikiPageWikiLink CW_complex.
- Complex_projective_space wikiPageWikiLink Category:Algebraic_varieties.
- Complex_projective_space wikiPageWikiLink Category:Complex_manifolds.
- Complex_projective_space wikiPageWikiLink Category:Projective_geometry.
- Complex_projective_space wikiPageWikiLink Cayley_plane.
- Complex_projective_space wikiPageWikiLink Characteristic_class.
- Complex_projective_space wikiPageWikiLink Characteristic_number.
- Complex_projective_space wikiPageWikiLink Chern_class.
- Complex_projective_space wikiPageWikiLink Circle.
- Complex_projective_space wikiPageWikiLink Circle_group.
- Complex_projective_space wikiPageWikiLink Classifying_space.
- Complex_projective_space wikiPageWikiLink Classifying_space_for_U(n).
- Complex_projective_space wikiPageWikiLink Cohomology.
- Complex_projective_space wikiPageWikiLink Cohomology_group.
- Complex_projective_space wikiPageWikiLink Commutative_ring.
- Complex_projective_space wikiPageWikiLink Compact_space.
- Complex_projective_space wikiPageWikiLink Complex_affine_space.
- Complex_projective_space wikiPageWikiLink Complex_manifold.
- Complex_projective_space wikiPageWikiLink Complex_number.
- Complex_projective_space wikiPageWikiLink Complex_plane.
- Complex_projective_space wikiPageWikiLink Complex_projective_plane.
- Complex_projective_space wikiPageWikiLink Cone_(linear_algebra).
- Complex_projective_space wikiPageWikiLink Connected_space.
- Complex_projective_space wikiPageWikiLink Cup_product.
- Complex_projective_space wikiPageWikiLink Cut_locus.
- Complex_projective_space wikiPageWikiLink Cut_locus_(Riemannian_manifold).
- Complex_projective_space wikiPageWikiLink Cyclic_group.
- Complex_projective_space wikiPageWikiLink Differentiable_manifold.
- Complex_projective_space wikiPageWikiLink Direct_limit.
- Complex_projective_space wikiPageWikiLink Divisor_(algebraic_geometry).
- Complex_projective_space wikiPageWikiLink Eilenberg–MacLane_space.
- Complex_projective_space wikiPageWikiLink Euclidean_space.
- Complex_projective_space wikiPageWikiLink Euler_characteristic.
- Complex_projective_space wikiPageWikiLink Foundations_of_Differential_Geometry.
- Complex_projective_space wikiPageWikiLink Fractional_linear_transformation.
- Complex_projective_space wikiPageWikiLink Fubini–Study_metric.
- Complex_projective_space wikiPageWikiLink Geodesic.
- Complex_projective_space wikiPageWikiLink Graded_ring.
- Complex_projective_space wikiPageWikiLink Grassmannian.
- Complex_projective_space wikiPageWikiLink Great_circle.
- Complex_projective_space wikiPageWikiLink Gromovs_inequality_for_complex_projective_space.
- Complex_projective_space wikiPageWikiLink Hermitian_symmetric_space.
- Complex_projective_space wikiPageWikiLink Hodge_number.
- Complex_projective_space wikiPageWikiLink Hodge_theory.
- Complex_projective_space wikiPageWikiLink Holomorphic_function.
- Complex_projective_space wikiPageWikiLink Homogeneous_coordinates.
- Complex_projective_space wikiPageWikiLink Homogeneous_function.
- Complex_projective_space wikiPageWikiLink Homogeneous_ideal.
- Complex_projective_space wikiPageWikiLink Homogeneous_space.
- Complex_projective_space wikiPageWikiLink Homology_(mathematics).
- Complex_projective_space wikiPageWikiLink Homology_group.
- Complex_projective_space wikiPageWikiLink Homotopy.
- Complex_projective_space wikiPageWikiLink Homotopy_group.
- Complex_projective_space wikiPageWikiLink Homotopy_theory.
- Complex_projective_space wikiPageWikiLink Hopf_bundle.
- Complex_projective_space wikiPageWikiLink Hopf_fibration.
- Complex_projective_space wikiPageWikiLink Hyperplane.
- Complex_projective_space wikiPageWikiLink Hyperplane_bundle.
- Complex_projective_space wikiPageWikiLink Ideal_(ring_theory).
- Complex_projective_space wikiPageWikiLink Inductive_limit.
- Complex_projective_space wikiPageWikiLink Infinite_cyclic.
- Complex_projective_space wikiPageWikiLink Italian_school_of_algebraic_geometry.
- Complex_projective_space wikiPageWikiLink John_Wiley_&_Sons.
- Complex_projective_space wikiPageWikiLink K(Z,2).
- Complex_projective_space wikiPageWikiLink Kähler_manifold.
- Complex_projective_space wikiPageWikiLink Kähler_metric.
- Complex_projective_space wikiPageWikiLink Lazare_Carnot.
- Complex_projective_space wikiPageWikiLink Lie_group.
- Complex_projective_space wikiPageWikiLink Line_at_infinity.
- Complex_projective_space wikiPageWikiLink Line_bundle.
- Complex_projective_space wikiPageWikiLink Linear_fractional_transformation.
- Complex_projective_space wikiPageWikiLink Local_ring.
- Complex_projective_space wikiPageWikiLink Localization_of_a_ring.
- Complex_projective_space wikiPageWikiLink Long_exact_homotopy_sequence.
- Complex_projective_space wikiPageWikiLink Mathematics.
- Complex_projective_space wikiPageWikiLink Meromorphic_function.
- Complex_projective_space wikiPageWikiLink Metric_tensor.
- Complex_projective_space wikiPageWikiLink N-sphere.
- Complex_projective_space wikiPageWikiLink Picard_group.
- Complex_projective_space wikiPageWikiLink Plane_at_infinity.
- Complex_projective_space wikiPageWikiLink Poincaré_duality.
- Complex_projective_space wikiPageWikiLink Polynomial.