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Old exam problems in MAT4510 can be found here:

http://www.uio.no/studier/emner/matnat/math/MAT4510/oppgaver/

The topic this week has been geodesics, section 5.6.  Again this was first developed for surfaces in Euclidean 3-space, but in the end we showed that the theory can be extended to any surface which is locally isometric to some such surface, since the equations can be expressed using only  the Riemannian metric. ("Being geodesic is an intrinsic property.")  Thus it also applies to the hyperbolic plane.  Among the most important examples, it is shown that the earlier notions of lines in Euclidean, spherical and hyperbolic geometry all are geodesic,  Moreover, an important result from differential equations implies that there is always exactly one geodesic line in every direction from every point.  Hence geodesics are indeed good generalizations of the concept of lines in a geometry.

Exercises for Tuesday November 18:  5.6:  1, 4-10

This week has been devoted to the important concept of Gaussian curvature, measuring how surfaces curve and bend in space.  The main theoretical result is Teorema Egregium, which says that curvature is invariant under isometries.  Next week we shall study the closest analogs of  lines in differential geometry, geodesics, and the rest of the semester we will explore the interactions between geodesics, curvature and the global topology of the surface.

Exercises for Tuesday, Nov 11:  5.4: 5, but replace the parametrization x(u,v) to  (u+cosh(v)i) and y(u,v) to the surface of rotation generated by the curve  (1/cosh(v), v-tanh(v)). (Same surface, but simpler calculations.)       5.5:  1, 2, 3.

Now the mandatory assignment has been graded.  Those of you who did not get your paper back in class, will find it on the shelves on the seventh floor of the Math building. Note that the deadline for submittting a second attempt is 1430, Thursday November 13.

This week we have covered the basic material needed for the study of differential geometry.  First the definition of tangent planes and derivatives of maps and functions in 5.1, and then the structure that all the remaining theory will depend on - the Riemannian metric  (5.3).  Finally we also introduced the maps that preserve the Riemannian structure, isometries.  Locally the metric can be described in terms of the inner products of basis vectors coming from local paramerizations, giving rise to three functions usually denoted E, F and G.  It is important that you become familiar with these from the beginning.   

Exercises for Tuesday November 4:  5.1: 3, 4,   5.3: 1, 3, 4.