Singular points of curve families on surfaces
摘要
l. Introdnction. It is well known that there are two cannonical bilinear forms on the tangent bundle of a smooth oriented surface which is immersed in the 3dimensional Euclidian space. These are called the first fundamental form and the second fundamental form. The principal curvature of the surface are defined by comparing these two forms. And a point where two principal curvatures coinside is called a umbilic point. Except for umblic points there exists a decomposition of the tangent space into two direct summands, each of which is tangent to the one of principal curvatures. And if a curve tangents t6 those tangent lines of direct summands at any points of it, it is called a curvature curve. In general for a given symmetric bilinear form on a 2-dimensional Riemannian manifold, we define the corresponding curve fammily with singularities and its local topological classification in the section 10. And we show the structurally stable condition of curve ･families on 2-dimensional closed Riemannian manifolds in the section 15.