- TOPOLOGY 385
Jacobi (see Klein, 1926, Vol. 2, p. 190), Beltrami arrived at the operatorwhere, with the notation slightly modernized, the Riemannian metric is given by
the usual ds^2 = o,ij dx* dx^, and ï denotes the determinant det(a,j). The
generalized operator is now referred to as the Laplace-Beltrami operator on a Rie-
mannian manifold.
The algebra of Grassmann and its connection with Riemann's general metric on
an ð-dimensional manifold was not fully codified until 1901, in "Methodes de calcul
differentiel absolu et leurs applications" ("Methods of absolute differential calculus
and their applications"), published in Mathematische Annalen in 1901, written by
Gregorio Ricci-Curbastro (1853-1925) and Tullio Levi-Civita (1873-1941). This
article contained the critical ideas of tensor analysis as it is now taught. The
absoluteness of the calculus consisted in the great generality of the transformations
that it permitted, showing how differential forms changed when coordinates were
changed. Although Ricci-Curbastro competed in a prize contest sponsored that year
by the Accademia dei Lincei, he was not successful, as some of the judges regarded
his absolute differential calculus as "useful but not essential"^40 to the development
of mathematics—the same sort of criticism leveled by Weierstrass against the work
of Hamilton in quaternions (see Section 2 of Chapter 15).
The following year Luigi Bianchi (1873-1928) published "Sui simboli a quattro
indice e sulla curvatura di Riemann" ("On the quadruply-indexed symbols and Rie-
mannian curvature"), in which he gave the relations among the covariant derivatives
of the Riemann curvature tensor, which, however, he derived by a direct method
for manifolds of constant curvature, not following the route of Ricci-Curbastro and
Levi-Civita. The Bianchi identity was later to play a crucial role in general relativ-
ity, assuring local conservation of energy when Einstein's gravitational equation is
assumed.4. Topology
Projections distort the shape of geometric objects, so that some metric properties
are lost. Some properties, such as parallelism, however, remain simply because the
number of intersections of two curves does not change. The study of space focusing
on such very general properties as connections and intersections has been known
by various names over the centuries. Latin has two words, locus and situs, meaning
roughly place and position. The word locus is one that we still use today to denote
the path followed by a point moving subject to stated constraints. It was the trans-
lation of the Greek word topos used by Pappus for the same concept. Since locus
was already in use, Leibniz fastened on situs and mentioned the need for a geom-
etry or analysis of situs in a 1679 letter to Huygens.^41 The meaning of geometria
situs and analysis situs evolved gradually. It seems to have been Johann Benedict
Listing (1808-1882) who, some time during the 1830s, realized that the Greek root
(^40) See the article on Ricci-Curbastro's paper at http: //www .math. unif i. it/matematicaitaliana/.
(^41) This letter was published in Huygens' (Euvres competes, M. Nijhoff, La Haye, 1888, Vol. 8,
p. 216. From the context it appears that Leibniz was calling for some simple way of expressing
position "as algebra expresses magnitude." If so, perhaps we now have what he wanted in the
form of vector analysis.