Projective connection

A differential-geometric structure on a smooth manifold $ M $; a special kind of connection on a manifold (cf. Connections on a manifold), where the smooth fibre space $ E $ over $ M $ has the projective space $ P _ {n} $ of dimension $ n = \mathop{\rm dim} M $ as its standard fibre. The structure of this $ E $ associates to each point $ x \in M $ a copy of the projective space, $ ( P _ {n} ) _ {x} $, which is identified (up to a homology with an invariant pencil of straight lines at the point $ x $) with the tangent centro-affine space $ T _ {x} ( M ) $ augmented by a hyperplane at infinity. As a connection in such an $ E $, a projective connection consists of the assignment of a projective mapping $ ( P _ {n} ) _ {x _ {t} } \rightarrow ( P _ {n} ) _ {x _ {0} } $ to each smooth curve $ {\mathcal L} \in M $ starting at $ x _ {0} $ and for each point $ x _ {t} $ of the curve in such a way that the following condition is satisfied. Let $ M $ be covered by coordinate regions in which the smooth field of frames in $ ( P _ {n} ) _ {x} $ is fixed, with the vertex determined by the vector $ e _ {0} $ coinciding with $ x $. (A frame in $ P _ {n} $ is determined by an equivalence class of bases in the vector space $ V _ {n+} 1 $, where two bases $ \{ e _ \alpha \} $ and $ \{ e _ \alpha ^ \prime \} $, $ \alpha = 0 \dots n $, are assumed to be equivalent if $ e _ \alpha ^ \prime = \lambda e _ \alpha $, $ \lambda \neq 0 $.) Then as $ t \rightarrow 0 $, the mapping in the family must tend to the identity mapping, and the principal part of its deviation from the identity mapping must be determined relative to the field of frames in some neighbourhood of the point $ x _ {0} $ by a matrix of linear differential forms

$$ \tag{1 } \omega _ \alpha ^ \beta = \Gamma _ {\beta i } ^ \alpha d x ^ {i} ,\ \mathop{\rm det} \| \Gamma _ {0i} ^ {j} \| \neq 0 , $$

$$ \alpha , \beta = 0 \dots n ; \ i , j = 1 \dots n, $$

common for all $ {\mathcal L} $. In other words, the image of the frame at a point $ x _ {t} $ under the mapping $ ( P _ {n} ) _ {x _ {t} } \rightarrow ( P _ {n} ) _ {x _ {0} } $ must be determined by the vectors

$$ e _ \beta [ \delta _ \alpha ^ \beta + \omega _ \alpha ^ \beta ( X ) t + \epsilon _ \alpha ^ \beta ( t ) ] , $$

where $ X $ is the tangent vector to $ {\mathcal L} $ at $ x _ {0} $ and $ \lim\limits _ {t \rightarrow 0 } \epsilon _ \alpha ^ \beta ( t ) / t = 0 $. The possibility of passing to equivalent bases leads to the fact that among the forms (1) only the forms

$$ \tag{2 } \omega _ {0} ^ {i} ,\ \theta _ {i} ^ {j} = \omega _ {i} ^ {j} - \delta _ {i} ^ {j} \omega _ {0} ^ {0} ,\ \omega _ {i} ^ {0} $$

are essential. When transforming the frame of the field at an arbitrary point $ x \in M $ according to the formulas $ e _ {\alpha ^ \prime } = A _ {\alpha ^ \prime } ^ \beta e _ \beta $, $ e _ \beta = A _ \beta ^ {\alpha ^ \prime } e _ {\alpha ^ \prime } $, where $ A _ {0 ^ \prime } ^ {i} = A _ {0} ^ {i ^ \prime } = 0 $, that is, when passing to an arbitrary element of the principal fibre space $ \Pi $ of frames in the spaces $ ( P _ {n} ) _ {x} $, the forms (1) are replaced by the following $ 1 $- forms on $ \Pi $:

$$ \tag{3 } \omega _ {\alpha ^ \prime } ^ {\beta ^ \prime } = A _ \gamma ^ { \beta ^ \prime } d A _ {\alpha ^ \prime } ^ \gamma + A _ {\alpha ^ \prime } ^ \gamma A _ \delta ^ {\beta ^ \prime } \omega _ \gamma ^ \delta . $$

The $ 2 $- forms

$$ \tag{4 } \Omega _ {\alpha ^ \prime } ^ {\beta ^ \prime } = d \omega _ {\alpha ^ \prime } ^ {\beta ^ \prime } + \omega _ {\gamma ^ \prime } ^ {\beta ^ \prime } \wedge \omega _ {\alpha ^ \prime } ^ {\gamma ^ \prime } $$

are semi-basic, that is, linear combinations of $ \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {1} $, and tensorial, that is, under the transformation of the frame by the matrices $ A _ \alpha ^ \gamma $ the formulas

$$ \Omega _ {\alpha ^ \prime } ^ {\beta ^ \prime } = A _ {\alpha ^ \prime } ^ \gamma A _ \delta ^ {\beta ^ \prime } \Omega _ \gamma ^ \delta $$

hold, where the $ \Omega _ {\alpha ^ \prime } ^ {\beta ^ \prime } $ are composed from (3) similarly to (4). For the essential forms (2) the structure equations of the projective connection hold (the primes are omitted for simplicity):

$$ \tag{5 } \left . \begin{array}{c} {d \omega _ {0} ^ {i} + \theta _ {j} ^ {i} \wedge \omega _ {0} ^ {j} = \ \Omega _ {0} ^ {i} , } \\ {d \theta _ {i} ^ {j} + \theta _ {k} ^ {j} \wedge \theta _ {i} ^ {k} + \omega _ {0} ^ {k} \wedge ( \delta _ {i} ^ {j} \omega _ {k} ^ {0} + \delta _ {k} ^ {j} \omega _ {i} ^ {0} ) = \Theta _ {i} ^ {j} , } \\ {d \omega _ {i} ^ {0} + \omega _ {k} ^ {0} \wedge \theta _ {i} ^ {k} = \ \Omega _ {i} ^ {0} , } \end{array} \right \} $$

where $ \Theta _ {i} ^ {j} = \Omega _ {i} ^ {j} - \delta _ {i} ^ {j} \Omega _ {0} ^ {0} $. Here, the right-hand sides are semi-basic; they constitute the system of torsion-curvature forms of the projective connection.

The equality $ \Omega _ {0} ^ {i} = 0 $ has an invariant meaning. In this case one speaks of a torsion-free projective connection; for it $ \Theta _ {i ^ \prime } ^ {i ^ \prime } = \Theta _ {i} ^ {i} $. The invariant identities

$$ \Omega _ {0} ^ {i} \equiv 0 ,\ \Theta _ {i} ^ {i} \equiv 0 , $$

$$ K _ {ikj} ^ {j} \equiv 0 ,\ \Theta _ {i} ^ {j} = \frac{1}{2} K _ {ikl} ^ {j} \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} $$

distinguish a special class of projective connections, called (by E. Cartan) normal projective connections.

The forms (1) determine a projective connection on $ M $ uniquely: The image, under the mapping $ ( P _ {n} ) _ {x _ {t} } \rightarrow ( P _ {n} ) _ {x _ {0} } $, of the frame at the point $ x _ {t} $ is determined by the solution $ \{ e _ \alpha ( t) \} $ of the system

$$ \tag{6 } du _ \alpha = ( \omega _ \alpha ^ \beta ) _ {x ( t ) } ( \dot{x} ( t ) ) u _ \beta $$

under the initial conditions $ u _ \alpha ( 0 ) = e _ \alpha $, where the $ x ^ {i} = x ^ {i} ( t ) $ are the equations of the curve $ {\mathcal L} $ in some coordinate neighbourhood of its point $ x _ {0} $ with coordinates $ x ^ {i} ( 0 ) $.

Any $ 1 $- forms $ \omega _ {0} ^ {i} , \theta _ {i} ^ {j} , \omega _ {i} ^ {0} $ defined on $ \Pi $ and satisfying equations (5) with right-hand sides expressible in terms of $ \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} $, where the $ \omega _ {0} ^ {i} $, $ i = 1 \dots n $, are linearly independent, define in this sense a projective connection on $ M $.

The curve described in $ ( P _ {n} ) _ {x _ {0} } $ by the point determined by the first vector $ e _ {0} ( t ) $ of the solution $ \{ e _ \alpha ( t ) \} $ of the system (6) is called the development of the curve $ {\mathcal L} $. A curve is called a geodesic line of the projective connection on $ M $ if its development in some neighbourhood of an arbitrary point $ x $ of it is a straight line of the space $ ( P _ {n} ) _ {x} $. The equations $ x ^ {i} = x ^ {i} ( t ) $ of a geodesic line are determined with the aid of the functions

$$ \xi ( t ) = ( \omega ^ {i} ) _ {x ( t ) } ( \dot{x} ( t ) ) $$

from the system

$$ d \xi ^ {i} + \xi ^ {j} ( \theta _ {j} ^ {i} ) _ {x ( t ) } ( \dot{x} ( t ) ) = \theta _ {x ( t ) } ( \dot{x} ( t ) ) \xi ^ {i} , $$

where $ \theta $ is a $ 1 $- form. In the frame where $ \omega ^ {i} = dx ^ {i} $ and $ \xi ^ {i} = \dot{x} ^ {i} $, this system has the form

$$ \tag{7 } \frac{d ^ {2} x ^ {a} }{( d x ^ {n} ) ^ {2} } = - Q ^ {a} \left ( \frac{d x ^ {1} }{d x ^ {n} } \dots \frac{d x ^ {n-} 1 }{d x ^ {n} } \right ) + $$

$$ + \frac{d x ^ {a} }{d x ^ {n} } Q ^ {n} \left ( \frac{d x ^ {1} }{d x ^ {n} } \dots \frac{d x ^ {n-} 1 }{d x ^ {n} } \right ) , $$

where $ Q ^ {a} $ and $ Q ^ {n} $ are polynomials of degree two with functions of $ x ^ {1} \dots x ^ {n} $ as coefficients.

Cartan's theorem: If on a smooth manifold $ M $ a system of curves is given that is locally defined by a system of differential equations of the form (7), then there is one and only one normal projective connection for which this system of curves is the system of geodesic lines.

The theory of projective connections thus supplies a means for an invariant investigation of systems of differential equations of a special form. Projective connections are also useful in the investigation of geodesic (or projective) mappings of spaces with affine connections. A projective connection reduces to an affine connection if on $ M $ there are local fields of frames with respect to which $ \omega _ {i} ^ {0} = P _ {ij} \omega _ {0} ^ {j} $. For every affine connection on $ M $ there is a unique normal projective connection with the same geodesic lines, from which the former can be obtained. Two affine connections are geodesically (or projectively) equivalent if their normal projective connections coincide. In particular, an affine connection on an $ M $ with $ \mathop{\rm dim} M > 2 $ is projectively Euclidean if and only if its projective curvature tensor $ K _ {ikl} ^ {j} $ vanishes.

References

[1]	E. Cartan, "Sur les variétés à connexion projective" Bull. Soc. Math. France , 52 (1924) pp. 205–241
[2]	E. Cartan, "Leçons sur la théorie des espaces à connexion projective" , Gauthier-Villars (1937)
[3]	S. Kobayashi, T. Nagano, "On projective connections" J. Math. and Mech. , 13 : 2 (1964) pp. 215–235
[a1]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)