The Schwarzian derivative for maps between manifolds with complex projective connections

Molzon, Robert; Mortensen, Karen Pinney

doi:10.1090/s0002-9947-96-01590-5

Cited by 37 publications

(19 citation statements)

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“…and apply to this the operator Dω. Then a further usage of (14) shows that the first summand vanishes so that we obtain…”

Section: 4mentioning

confidence: 97%

“…We refer the reader to [13] and [17] for surveys of this classical theory. Both pre-Schwarzian and Schwarzian derivatives have been generalized and studied in the settings of harmonic mappings in the plane (see [3,11]) and of holomorphic mappings in C n (see [19] and [9,14,16,18], for example). The main purpose of this article is to extend both operators to pluriharmonic mappings in C n .…”

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confidence: 99%

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Schwarzian derivatives for pluriharmonic mappings

Efraimidis¹,

Ferrada-Salas²,

Hernández³

et al. 2019

Preprint

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A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in C n are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Möbius transformation.Some observations are made related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations, revealing differences between the theories in the plane and in higher dimensions. An example is given that rules out the possibility for a shear construction theorem to hold in C n , for n ≥ 2.

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“…and apply to this the operator Dω. Then a further usage of (14) shows that the first summand vanishes so that we obtain…”

Section: 4mentioning

confidence: 97%

mentioning

confidence: 99%

Schwarzian derivatives for pluriharmonic mappings

Efraimidis¹,

Ferrada-Salas²,

Hernández³

et al. 2019

Preprint

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“…The following definition was motivated by R. Molzon and K. Pinney Mortensen [15]. Let M be a complex manifold of dimension n ≥ 2.…”

Section: Riccati Connectionsmentioning

confidence: 99%

Affine connections on complex compact surfaces and Riccati distributions

Lizarbe¹

2021

Preprint

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“…The multi-dimensional Schwarzian derivative introduced in this section has been around for quite a while, see, e.g., [73,124,234,155]; most of these references deal with the complex analytic case. Theorem 7.1.6 is new.…”

Section: Commentmentioning

confidence: 99%

Projective Differential Geometry Old and New

Ovsienko

Tabachnikov

2004

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Metrical geometry is a part of descriptive geometry 1 , and descriptive geometry is all geometry. Arthur Cayley On October 5-th 2001, the authors of this book typed in the word "Schwarzian" in the MathSciNet database and the system returned 666 hits. Every working mathematician has encountered the Schwarzian derivative at some point of his education and, most likely, tried to forget this rather scary expression right away. One of the goals of this book is to convince the reader that the Schwarzian derivative is neither complicated nor exotic, in fact, this is a beautiful and natural geometrical object. The Schwarzian derivative was discovered by Lagrange: "According to a communication for which I am indebted to Herr Schwarz, this expression occurs in Lagrange's researches on conformable representation 'Sur la construction des cartes géographiques' " [117]; the Schwarzian also appeared in a paper by Kummer in 1836, and it was named after Schwarz by Cayley. The main two sources of current publications involving this notion are classical complex analysis and one-dimensional dynamics. In modern mathematical physics, the Schwarzian derivative is mostly associated with conformal field theory. It also remains a source of inspiration for geometers. The Schwarzian derivative is the simplest projective differential invariant, namely, an invariant of a real projective line diffeomorphism under the natural SL(2, R)-action on RP 1. The unavoidable complexity of the formula for the Schwarzian is due to the fact that SL(2, R) is so large a group (three-dimensional symmetry group of a one-dimensional space). Projective geometry is simpler than affine or Euclidean ones: in projective geometry, there are no parallel lines or right angles, and all nondegenerate conics are equivalent. This shortage of projective invariants is This curve is non-degenerate at point γ(0). The plane, dual to point γ(t), is given, in an appropriate affine coordinate system (a 1 , a 2 , a 3) in RP 3 * , by 1.2. DISCRETE INVARIANTS AND CONFIGURATIONS 1.5. SCHWARZIAN DERIVATIVE AS A COCYCLE OF DIFF(RP 1) 19 counterpart of the Huygens construction of the involute of a plane curve using a non-stretchable string: the role of the tangent line is played by the osculating conic and the role of the Euclidean length by the projective one. Affine differential geometry and the corresponding differential invariants appeared later than the projective ones. A systematic theory was developed between 1910 and 1930, mostly by Blaschke's school. 1.5 Schwarzian derivative as a cocycle of Diff(RP 1) The oldest differential invariant of projective geometry, the Schwarzian derivative, remains the most interesting one. In this section we switch gears and discuss the relation of the Schwarzian derivative with cohomology of the group Diff(RP 1). This contemporary viewpoint leads to promising applications that will be discussed later in the book. To better understand the material of this and the next section, the reader is recommended to consult Section 8.4. Invariant and relative ...

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The Schwarzian derivative for maps between manifolds with complex projective connections

Cited by 37 publications

References 19 publications

Schwarzian derivatives for pluriharmonic mappings

Schwarzian derivatives for pluriharmonic mappings

Affine connections on complex compact surfaces and Riccati distributions

Projective Differential Geometry Old and New

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