401-3531-00L  Differential Geometry I

SemesterAutumn Semester 2016
LecturersU. Lang
Periodicityyearly recurring course
Language of instructionGerman
CommentThis course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from neither 401-3531-00L Differential Geometry I nor 401-3532-00L Differential Geometry II for their Bachelor's degree still can have recognised this course for the Master's degree.
Furthermore, at most one of the three course units
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics.


401-3531-00 VDifferentialgeometrie I4 hrs
Tue10-12HG E 7 »
Thu10-12HG G 5 »
U. Lang
401-3531-00 UDifferentialgeometrie I
Do 13-14 oder Do 14-15 oder Fr 13-14
1 hrs
Thu13-14CAB G 52 »
14-15HG E 21 »
14-15ML H 41.1 »
Fri13-14HG G 26.3 »
U. Lang

Catalogue data

AbstractCurves in R^n, inner geometry of hypersurfaces in R^n, curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, tangent bundle, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
ObjectiveIntroduction to elementary differential geometry and differential topology.
Content- Differential geometry in R^n: theory of curves, submanifolds and immersions, inner geometry of hypersurfaces, Gauss map and curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet, Poincaré Index Theorem.
- The hyperbolic space.
- Differential topology: differentiable manifolds, tangent bundle, immersions and embeddings in R^n, Sard's Theorem, transversality, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
LiteratureDifferential Geometry in R^n:
- Manfredo P. do Carmo: Differential geometry of curves and surfaces
- Wolfgang Kühnel: Differentialgeometrie. Curves-surfaces-manifolds
- Christian Bär: Elementary differential geometry
Differential Topology:
- Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds
- Victor Guillemin & Alan Pollack: Differential Topology
- Morris W. Hirsch: Differential Topology

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits10 credits
ExaminersU. Lang
Typesession examination
Language of examinationGerman
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examinationoral 30 minutes
Additional information on mode of examinationPrüfungssprache: Deutsch oder Englisch / Language of examination: English or German
This information can be updated until the beginning of the semester; information on the examination timetable is binding.

Learning materials

Teaching document repositoryLehr-Dokumentenablage / Teaching document repository
Only public learning materials are listed.


No information on groups available.


There are no additional restrictions for the registration.

Offered in

High-Energy Physics (Joint Master with EP Paris)Optional Subjects in MathematicsWInformation
Mathematics BachelorCore Courses: Pure MathematicsWInformation
Mathematics Master(also Bachelor) Core Courses: Pure MathematicsWInformation
Physics BachelorSelection of Higher Semester CoursesWInformation
Physics MasterSelection: MathematicsWInformation