SUBJECT
Title
Differential geometry I
Type of instruction
lecture + practical
Level
master
Faculty
Part of degree program
Credits
2+3
Recommended in
Semesters 1-4
Typically offered in
Autumn/Spring semester
Course description
Smooth parameterized curves in the n-dimensional Euclidean space Rn. Arc length parameterization. Distinguished Frenet frame. Curvature functions, Frenet formulas. Fundamental theorem of the theory of curves. Signed curvature of a plane curve. Four vertex theorem. Theorems on total curvatures of closed curves.
Smooth hypersurfaces in Rn. Parameterizations. Tangent space at a point. First fundamental form. Normal curvature, Meusnier’s theorem. Weingarten mapping, principal curvatures and directions. Christoffel symbols. Compatibility equations. Theorema egregium. Fundamental theorem of the local theory of hypersurfaces. Geodesic curves.
Readings
- M. P. do Carmo: Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, 1976.
Further reading:
- B. O’Neill: Elementary differential geometry. Academic Press, New York, 1966.