chapter  6
28 Pages

Two-Dimensional Surfaces in £ 4

In this section we will study in a more detailed way the properties of two-dimensional surfaces in four-dimensional Euclidean space.

/1 12/i dui = (11 I 112u') dui.

To each point p E F2 and each direction T E Tp there corresponds the vector kN(T) of normal curvature in the normal space NP . Put the initial point of kN( T) top. Then its end-point is the point M of the normal curvature indicatrix. If T rotates in the tangent plane Tp then M traces some closed curve. For the arbitrary direction T of infinitesimal components (du 1, du2 ) we have

Then rp is the angle between T and the u1 coordinate curve. For the coordinates of indicatrix points we obtain the expression

xi = L~ 1 cos2 rp + 2Lb cos rp sin rp + L~2 sin2 rp. Transform the expression above to the form

Set

Since giJ = 15u at p, in our system a and (3 are the components of mean curvature vector H, i.e.