## MATH 433 - DIFFERENTIAL GEOMETRY I

## Fall 1998:

## EXERCISES

** Chapter. I **

** 18 September 1998**

You are expected to solve all exercises at the end of each sections
of Manfredo do Carmo. The following problems are from the Eisenharts book.
The binormal at a point M of a curve is the limiting position of the
common perpendicular to tangents at M and M' , as M' approaches M.
Find a necessary and sufificient condition that a curve lie upon
a sphere.
Find the intrinsic and paramteric equations of a plane curve which
is such that the segment on any tangent between the point of contact and the
projection of a fixed point is of constant length.
Find the intrinsic equation of the plane curve which meets under the constant angle all the lines passing through a fixed point.
Investigate the curve which is the locus of the point on the principal normal of a given curve and at constant distance from the later.
Find the vector representing the rate of change of the acceleration of a
moving point.

## Last update September 1998