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Projective connection

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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.

Projective space as the model geometry

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 M {\displaystyle M} of the homogeneous space is the projective space RP which we shall represent by homogeneous coordinates [ x 0 , … , x n ] {\displaystyle } . The symmetry group of M {\displaystyle M} is G = PSL(n+1,R). Let H be the isotropy group of the point [ 1 , 0 , 0 , … , 0 ] {\displaystyle } . Thus, M = G/H presents M {\displaystyle M} as a homogeneous space.

Let g {\displaystyle {\mathfrak {g}}} be the Lie algebra of G, and h {\displaystyle {\mathfrak {h}}} that of H. Note that g = s l ( n + 1 , R ) {\displaystyle {\mathfrak {g}}={\mathfrak {s}}{\mathfrak {l}}(n+1,{\mathbb {R} })} . As matrices relative to the homogeneous basis, g {\displaystyle {\mathfrak {g}}} consists of trace-free ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1)\times (n+1)} matrices:

( λ v i w j a j i ) , ( v i ) ∈ R 1 × n , ( w j ) ∈ R n × 1 , ( a j i ) ∈ R n × n , λ = − ∑ i a i i {\displaystyle \left({\begin{matrix}\lambda &v^{i}\\w_{j}&a_{j}^{i}\end{matrix}}\right),\quad (v^{i})\in {\mathbb {R} }^{1\times n},(w_{j})\in {\mathbb {R} }^{n\times 1},(a_{j}^{i})\in {\mathbb {R} }^{n\times n},\lambda =-\sum _{i}a_{i}^{i}} .

And h {\displaystyle {\mathfrak {h}}} consists of all these matrices with ( w j ) = 0 {\displaystyle (w_{j})=0} . Relative to the matrix representation above, the Maurer-Cartan form of G is a system of 1-forms ( ξ , α j , α j i , α i ) {\displaystyle (\xi ,\alpha _{j},\alpha _{j}^{i},\alpha ^{i})} satisfying the structural equations (written using the Einstein summation convention):

d ξ + α i ∧ α i = 0 {\displaystyle d\xi +\alpha ^{i}\wedge \alpha _{i}=0}
d a j + a j ∧ ζ + a j k ∧ a k = 0 {\displaystyle da_{j}+a_{j}\wedge \zeta +a_{j}^{k}\wedge a_{k}=0}
d a j i + a i ∧ a j + a k i ∧ a j k = 0 {\displaystyle da_{j}^{i}+a^{i}\wedge a_{j}+a_{k}^{i}\wedge a_{j}^{k}=0}
d a i + ζ ∧ a i + a k ∧ a k i = 0 {\displaystyle da^{i}+\zeta \wedge a^{i}+a^{k}\wedge a_{k}^{i}=0}

Projective structures on manifolds

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),

Une variété (ou espace) à connexion projective est une variété numérique qui, au voisinage immédiat de chaque point, présente tous les caractères d'un espace projectif et douée de plus d'une loi permettant de raccorder en un seul espace projectif les deux petits morceaux qui entourent deux points infiniment voisins. ...
Analytiquement, on choisira, d'une manière d'ailleurs arbitraire, dans l'espace projectif attaché à chaque point a de la variété, un repére définissant un système de coordonnées projectives. ... Le raccord entre les espaces projectifs attachés à deux points infiniment voisins a et a' se traduira analytiquement par une transformation homographique. ...

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):

La variété sera dite à "connexion affine" lorsqu'on aura défini, d'une manière d'ailleurs arbitraire, une loi permettant de repérer l'un par rapport à l'autre les espaces affines attachés à deux points infiniment voisins quelconques m et m' de la variété; cete loi permettra de dire que tel point de l'espace affine attaché au point m' correspond à tel point de l'espace affine attaché au point m, que tel vecteur du premier espace es parallèle ou équipollent à tel vecteur du second espace.

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.

Notes

  1. It is also possible to use PGL(n+1,R), but PSL(n+1,R) is more convenient because it is connected.
  2. Cartan's approach was to derive the structural equations from the volume-preserving condition on SL(n+1) so that explicit reference to the Lie algebra was not required.
  3. A point of interest is this last equation is completely integrable, which means that the fibres of G → G / H {\displaystyle G\rightarrow G/H} can be defined using only the Maurer-Cartan form, by the Frobenius integration theorem.
  4. A variety (or space) with projective connection is a numerical variety which, in the immediate neighbourhood of each point, possesses all the characters of a projective space and is moreover endowed with a law making it possible to connect in a single projective space the two small regions which surround two infinitely close points. Analytically, we choose, in a way otherwise arbitrary, a frame defining a projective frame of reference in the projective space attached to each point of the variety. .. The connection between the projective spaces attached to two infinitely close points a and a' will result analytically in a homographic (projective) transformation. ..
  5. The variety will be said to "affinely connected" when one defines, in a way otherwise arbitrary, a law making it possible to place the affine spaces, attached to two arbitrary infinitely close points m and m' of the variety, in correspondence with each other; this law will make it possible to say that a particular point of the affine space attached to the point m' corresponds to a particular point of the affine space attached to the point m, in such a way that a vector of the first space is parallel or equipollent with the corresponding vector of the second space.

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