Geoscience Reference
In-Depth Information
Fig. 7 The concept of ramification. Here the ramification points of the union of the two cylinders
is the common intersection curve of these two cylinders. The neighborhood (hashed ball without
its bounding sphere) of a ramification point of the union of two cylinders contains unramified
points on two different sheets—the two cylinders without their boundaries and common
intersection—, whose projection onto the plane are contained in a single neighborhood (hashed
disk without its bounding circle) of the projection of the ramification point onto the plane. The
degree of the covering map (projection in this case) is two, since each point on the neighborhood
of the projection of the ramification point onto the plane has two antecedents by the covering
map: one on each sheet, except at the ramification points, where the two sheets meet
the Hurwitz theorem in algebraic topology, which is a special case of the Riemann-
Roch theorem in algebraic geometry.
Each point p of a geometric object X of dimension d has a neighborhood V
ðÞ
which is ''evenly covered'' by sheets i.e., disjoint arc-connected open sets (where
each point of these sheets has a neighborhood homeomorphic to a ball without its
bounding sphere and it is possible to walk from any point to any other point of
these sheets without leaving them, see Fig.
7
). The minimum number of sheets
(path-connected disjoint open sets) needed to cover X at p is referred to as the
number of sheets N
p
around p, and we say that the covering of the geometric object
is N
p
-sheeted around p, or the degree of the covering map (mapping the geometric
object to a d
1 dimensional space like the projection onto the plane in Fig.
7
)is
N
p
around p. Intuitively, a ramification point q is a point where several sheets meet
together, i.e., the number of sheets is larger than one in all the points of the
neighborhood of q except at q (where the sheets meet, see Fig.
7
, where the ram-
ification points are the points on the intersection of the two cylinders, the number of
sheets at these points is one, while the number of sheets is two in their neighborhood
