384 12. MODERN GEOMETRIESwhether a surface can be synthesized from any six functions regarded as the coef-
ficients of these forms. Do they determine the surface, up to the usual Euclidean
motions of translation, rotation, and reflection that can be used to move any pre-
scribed point to a prescribed position and orientation? Such a theorem does hold for
curves, as was established by two French mathematicians, Jean Frenet (1816-1900)
and Joseph Serret (1818-1885), who gave a set of equations—the Frenet-Serret^37
equations—determining the curvature and torsion of a curve in three-dimensional
spaces. A curve can be reconstructed from its curvature and torsion up to trans-
lation, rotation, and reflection. A natural related question is: Which sets of six
functions, regarded as the components of the two fundamental forms, can be used
to construct a surface? After all, one needs generally only three functions of two
parameters to determine the surface, so that the six given by Gauss cannot be
independent of one another.
In an 1856 paper, Gaspare Mainardi (1800-1879) provided consistency condi-
tions in the form of four differential equations, now known as the Mainardi-Codazzi
equations,^38 that must be satisfied by the six functions E, F, G, D, D', and D" if
they are to be the components of the first and second fundamental forms introduced
by Gauss. Mainardi had learned of Gauss' work through a French translation, which
had appeared in 1852. These same equations were discovered by Delfino Codazzi
(1824-1875) two years later, using an entirely different approach, and helped him to
win a prize from the Paris Academy of Sciences. Codazzi published these equations
only in 1883.
When Riemann's lecture was published in 1867, the year after his death, it
became the point of departure for a great deal of research in Italy.^39 One who
worked to develop these ideas was Riemann's friend Enrico Betti (1823-1892), who
tried to get Riemann a chair of mathematics in Palermo. These ideas led Betti
to the notion of the connectivity of a surface. On the simplest surfaces, such as a
sphere, every closed curve is the boundary of a region. On a torus, however, the
circles of latitude and longitude are not boundaries. These ideas belong properly
to topology, discussed in the next section. In his fundamental work on this subject,
Henri Poincare named the maximum number of independent non-boundary cycles
in a surface the Betti number of the surface, a concept that is now generalized to
ç dimensions. The nth Betti number is the rank of the nth homology group.
Another Italian mathematician who extended Riemann's ideas was Eugenio
Beltrami (1835-1900), whose 1868 paper on spaces of constant curvature contained
a model of a three-dimensional space of constant negative curvature. Beltrami had
previously given the model of a pseudosphere, as explained in Chapter 11, to repre-
sent the hyperbolic plane. It was not obvious before his work that three-dimensional
hyperbolic geometry and a three-dimensional manifold of constant negative curva-
ture were basically the same thing. Beltrami also worked out the appropriate n-
dimensional analogue of the Laplacian ^ + + §pr, which plays a fundamental
role in mathematical physics. By working with an integral considered earlier by
(^37) Frenet gave six equations for the direction cosines of the tangent and principal normal to the
curve and its radius of curvature. Serret gave the full set of nine now called by this name, which
are more symmetric but contain no more information than the six of Frenet.
(^38) The Latvian mathematician Karl Mikhailovich Peterson (1828-1881) published an equivalent
set of equations in Moscow in 1853, but they went unnoticed for a full century.
(^39) Riemann went to Italy for his health and died of tuberculosis in Selasca. He was in close contact
with Italian mathematicians and even published a paper in Italian.