DIFFERENTIAL GEOMETRY: MATH 3113 and ADVANCED ...

DIFFERENTIAL GEOMETRY: MATH 3113 and
ADVANCED DIFFERENTIAL GEOMETRY:
MATH 5113
Course Information
Where and when:
• Tuesdays at 10 (Chemistry G.74) and Thursdays at 12 (RSLT
19) - these are lectures
• Fridays at 11 (RSLT 19) - 4 of 5 of these will be lectures and
the remainder example classes
Module webpage:
http://www.maths.leeds.ac.uk/ rb/DiffGeom34.htm
Reading list:
[1] B. O’Neill, Elementary differential geometry, Academic Press, 1966, 2nd ed.
1997
[2] J. McCleary, Geometry from a differentiable viewpoint, C.U.P., 1994.
[3] J. Oprea, Differential geometry and its applications, Prentice Hall, 2004.
[4] M. Do Carmo, Differential geometry of curves and surfaces, Prentice Hall,
1976.
[5] J.A. Thorpe, Elementary topics in differential geometry, Springer, 1979.
[6] C. Bär, Elementary differential geometry, C.U.P., 2010.
Short description:
We are interested in global properties of curves and surfaces and the
relation between local quantities and global invariants.
In the first part of the course, we meet the Whitney-Graustein theorem,
which says that two curves can be deformed into each other
as soon as a single number (the rotation index) is the same for both
curves. We give some applications of this result, before proceeding
to study global properties of n-surfaces, such as isometries, shortest
curves etc. We meet the Theorema Egregium of Gauss which says that
the Gauss curvature of a surface is intrinsic, we contrast this with the
mean curvature, which is zero for a soap film, but depends crucially on
how that soap film lies in 3-space.
More generally, we examine what properties are preserved by transformations,
with applications to map projections of the surface of the
earth. We finish with the celebrated Gauss-Bonnet theorem, which
says that the total curvature of a surface is unchanged however much
the surface is deformed, for example for any surface which ”looks like”
a sphere, it is 4π.
A common theme is that of ”curvature”, this concept underpins
much modern maths, for example, the curved universe of general relativity
theory.
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Topics include:
• Plane curves: rotation index, Whitney-Graustein theorem, turning
tangent theorem
• Space curves: congruence
• Submanifolds of Euclidean spaces as level sets
• Gauss map
• Curves on a surface, geodesics
• Transformations and map projections
• Theorema Egregrium
• Gauss-Bonnet theorem
ADVANCED DIFFERENTIAL GEOMETRY:
MATH 5113 only
Wednesdays at 11 (RSLT 10) there will be an additional lecture.
The webpage and the reading list is the same as for MATH3113.
As for the course description, several topics will be taken further than
Differential Geometry, and an understanding of basic ideas of abstract
Riemannian Geometry will be gained.
Topics include:
• Plane curves: rotation index, isoperimetric inequality, Fenchel’s
theorem.
• Space curves: congruence, total curvature of a knot.
• Submanifolds of Euclidean spaces as level sets, Gauss map.
• Curves on a surface, geodesics.
• Gauss Lemma and a proof that geodesics minimise distance
locally.
• Isometries and conformal maps.
• Theorema Egregrium. Gauss-Bonnet theorem. Abstract Riemannian
metrics.