Universität Wien
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250024 VO Differential Geometry (2026S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine (iCal) - nächster Termin ist mit N markiert

  • Montag 02.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 04.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 09.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 11.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 16.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 18.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 23.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 25.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 13.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 15.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 20.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 22.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 27.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 29.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 04.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 06.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 11.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 13.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 18.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 20.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 27.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 01.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 03.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 08.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 10.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 15.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 17.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 22.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Mittwoch 24.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 29.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

Prerequisites

I expect students to have very good working knowledge of multivariable calculus (including the chain rule, the inverse function theorem, and the transformation formula), point set topology (definition of a topological space, induced topology, connectedness, compactness, Hausdorff property, continuity; metric spaces, completeness, Banach fixed point theorem; Euclidean space and its topological properties), and linear algebra (including the matrix representation of linear maps; trace, determinant, eigenvalues, eigenvectors, and diagonalizability of automorphisms; dual vector space and dual of linear maps; inner product spaces, orthonormal systems, Gram-Schmidt process).

Syllabus

Review of prerequisites and applications

Analysis in Euclidean space: partition of unity subordinate to an open cover; differential forms (wedge product, exterior derivative, pull-back, closed and exact forms); tangent fields (action on smooth functions, Lie bracket), Cartan’s magic formula; invariance under diffeomorphisms; smooth functions on half-spaces; divergence theorem for compact domains; possibly Sard’s theorem

Curves in Euclidean space: length and reparametrization of curves; special reparametrizations of regular curves; geometry of plane curves (tangent field, normal field, curvature, total curvature, fundamental theorem of plane curves, osculating circles, angle functions, winding number and rotation number of periodic curves; homotopy invariance; Hopf’s Umlaufsatz)

Submanifolds of Euclidean space: submanifolds with and without boundary; submanifolds from equations; tangent space; derivative of functions on submanifolds in direction of tangent vectors; integration of functions; orientability; integration of differential forms; Stokes’ theorem; classical integration theorems

Geometry of immersions into Euclidean space: principles of map-mapping, first fundamental form; tangential and normal projection; second fundamental form; Gauss equation; Codazzi equation; immersions of co-dimension one (Gauss map, principal curvatures, Theorema Egregium); reparametrization (geometric invariance of area, first and second fundamental form, curvature; local graph parametrization; uniform local graph parametrization); the geodesic equation; geodesic coordinate systems; possibly existence of reparamatrization to geodesic coordinates

Differential topology: proof that closed differential forms on star-shaped domains are exact (Poincaré lemma); Brouwer’s fixed point theorem; hedgehog theorem; possibly the Gauss-Bonnet theorem; possibly the Jordan curve theorem

Abstract manifolds: calculus on manifolds (smooth functions, smooth maps, immersions, submersions, diffeomorphisms, partitions of unity, tangent fields, differential forms, tensor fields); gluing theorem; Whitney embedding theorem; Stokes’ theorem; possibly elements of de Rham cohomology

I will provide terse lecture notes for all topics except the review of prerequisites. Please attend the lecture course for additional details required, e.g., in the exam.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Your grade will be based on a written 90-minute exam.
You are only allowed to bring writing utensils to the exam.

Mindestanforderungen und Beurteilungsmaßstab

You need to achieve at least half the marks on the final exam to pass this course.
If your percentage mark on the exam is p, then you will be awarded the following grade:
Sehr gut (1) for p in [87.5; 100]
Gut (2) for p in [75; 87.5)
Befriedigend (3) for p in [62.5; 75)
Genügend (4) for p in [50; 62.5)
Nicht genügend (5) for p in [0; 50)

Prüfungsstoff

All content discussed in class is examinable, unless explicitly declared otherwise.
In the exam, you will also be asked to apply the theory from class to problems of a similar complexity as those on the weekly problems sets.

Literatur

Zuordnung im Vorlesungsverzeichnis

MGED; ML1; MEL

Letzte Änderung: Fr 10.07.2026 15:26