By Ovsienko V., Tabachnikov S.

Principles of projective geometry maintain reappearing in possible unrelated fields of arithmetic. This e-book offers a fast direction for graduate scholars and researchers to consider the frontiers of up to date examine during this vintage topic. The authors comprise routines and historic and cultural reviews bearing on the fundamental rules to a broader context.

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1 Invariant differential operators on RP1 The language of invariant differential operators is an adequate language for differential geometry. The best-known invariant differential operators are the de Rham differential of differential forms and the commutator of vector ﬁelds. These operators are invariant with respect to the action of the group of diffeomorphisms of the manifold. The expressions that describe these operations are independent of the choice of local coordinates. If a manifold M carries a geometric structure, the notion of the invariant differential operator changes accordingly: the full group of diffeomorphisms is restricted to the groups preserving the geometric structure.

22). 46 2 The geometry of the projective line As a consequence of the previous constructions, we obtain higher differential invariants of non-degenerate curves in RPn . 4. The choice of a projective structure allows us to reduce the inﬁnite-dimensional group Diff(S 1 ) to PGL(2, R). 7). 22) is a projective differential invariant of the curve. 4, is actually invariant under the full group Diff(S 1 ). The constructed set of projective invariants is complete. Indeed, the differential operator A characterizes the curve up to projective equivalence, while the total symbol completely determines the operator.

Thus one has an action of Diff(S 1 ) on the space of such operators. We call it the geometric action. 2 Curves in RP n and linear differential operators 31 which is, of course, a particular case of the action of Diff(S 1 ) on Dλ,μ (S 1 ) as n+1 (S 1 ). 1). 7), namely, the highest-order coefﬁcient equals 1 and the next highest equals zero, if and only if n+2 n and μ= . 7) coincide. Proof Let us start with the geometric action. Consider a new parameter y = f (x) on γ . , where = y f , xx = yy ( f )2 + y f , is a lifted curve and f denotes d f /dx.