  A Laplacian eigenfunction on a two-dimensional manifold dictates some natural
partitions of the manifold; the most apparent one being the well studied nodal
domain partition. An alternative partition is revealed by considering a set of
distinguished gradient flow lines of the eigenfunction - those which are
connected to saddle points. These give rise to Neumann domains. We establish
complementary definitions for Neumann domains and Neumann lines and use basic
Morse homology to prove their fundamental topological properties. We study the
eigenfunction restrictions to these domains. Their zero set, critical points
and spectral properties allow to discuss some aspects of counting the number of
Neumann domains and estimating their geometry.
