 
Summary: REMARKS ON THE PARABOLIC CURVES ON SURFACES AND
ON THE MANYDIMENSIONAL M ¨
OBIUSSTURM THEORY
V. I. Arnold
V. A. Steklov Mathematical Institute
August 26, 1997
Conjecture on Four Parabolic Curves
According to the classical M¨obius theorem, a curve on the projective plane has at
least three inflection points provided that this curve is close to a projective line. This
theorem is a special case of the SturmHurwitz theorem [2] stating that the number of
the sign alternations of a function on a circle is not less than the number of the sign
alternation of the lowest harmonic entering its Fourier series with nonzero coefficient.
The M¨obius theorem can be extended to all noncontractible curves embedded in the
projective plane. The Sturm theorem has probably a global version too. Attempts to
extend the M¨obius and Sturm theorems to the manydimensional case (i.e., to surfaces
in the projective space and to functions of several variables) show that natural analogs of
inflection points are parabolic curves of surfaces and zeros of the Hessians of functions.
This means that the differential operators corresponding to surfaces and to functions
of two variables are quadratic (of degree n in the case of ndimensional manifolds and
functions of n variables) rather than linear (as is the case in the ordinary Sturm theory
