Differential Geometry. Prerequiste: MATH 224. This course studies the geometry
of curves and surfaces in Euclidean space: intrinsic and extrinsic geometry, curvature,
structural equations of Riemannian geometry, and the Gauss-Bonnet theorem. Prerequisites
are Math 224 and a bit of mathematical sophistication.
Text: "Differential Geometry of Curves and Surfaces",
Revised and Updated Second Edtion, by Manfredo Do Carmo, Noew York, Dover Publications.
ISBN:-13 978-0-48-680699-0.
Office
Hours. I will have office hours on Monday and Wednesday in person
and will be available for Zoom hours on Tuesday and Thursday.. You are always
welcome!
The geometry of curves in the plane and in (Euclidean) space. Arclength,
curvature, torsion, Frenet equations, some interesting examples, problems
about curves. Existence and rigidity theorem for curves in the plane and space
curves.
Surfaces, tangent plane, normals, first fundamental form (Riemannian metric),
Covariant derivative and parallelism, second fundamental form (curvature),
examples. Curves on surfaces.
Gaussian curvature and Mean curvature. The fundamental equations of surface
theory.
Geodesics on surfaces.
Intrinsic geometry of surfaces. The Gauss-Bonnet theorm. Theorem of Hopf
and Rinow.
Topics in differential geometry as time permits..
Course Information:(DATES MAY CHANGE) There will be regular
homework assignments, which will count for 30% of the grade. . There will be
a midterm exam in class (tentatively) scheduled for Wednesday, March
5, 2025, which will count for 30%. The final exam will count for the
remaining 40%.
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