In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.
The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections, projective connections also define geodesics. However, these geodesics are not affinely parametrized. Rather they are projectively parametrized, meaning that their preferred class of parameterizations is acted upon by the group of fractional linear transformations.
Like an affine connection, projective connections have associated torsion and curvature.
The first step in defining any Cartan connection is to consider the flat case: in which the connection corresponds to the Maurer-Cartan form on a homogeneous space.
In the projective setting, the underlying manifold of the homogeneous space is the projective space RPn which we shall represent by homogeneous coordinates . The symmetry group of is G = PSL(n+1,R). [1] Let H be the isotropy group of the point . Thus, M = G/H presents as a homogeneous space.
Let be the Lie algebra of G, and that of H. Note that . As matrices relative to the homogeneous basis, consists of trace-free matrices:
And consists of all these matrices with . Relative to the matrix representation above, the Maurer-Cartan form of G is a system of 1-forms satisfying the structural equations (written using the Einstein summation convention): [2]
A projective structure is a linear geometry on a manifold in which two nearby points are connected by a line (i.e., an unparametrized geodesic) in a unique manner. Furthermore, an infinitesimal neighborhood of each point is equipped with a class of projective frames . According to Cartan (1924),
This is analogous to Cartan's notion of an affine connection , in which nearby points are thus connected and have an affine frame of reference which is transported from one to the other (Cartan, 1923):
In modern language, a projective structure on an n-manifold M is a Cartan geometry modelled on projective space, where the latter is viewed as a homogeneous space for PSL(n+1,R). In other words it is a PSL(n+1,R)-bundle equipped with
such that the solder form induced by these data is an isomorphism.
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In mathematics, Hilbert's Nullstellensatz is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893.
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