Program Overview

---

Program Details

Course Overview

The course Geometry 2 (Geom2) is a single subject course that covers various topics in geometry, including differentiable manifolds, tangent spaces, and differential forms.

Course Content

The following subjects are covered in the course:

• Differentiable manifolds in Euclidean spaces
• Abstract differentiable manifolds
• Tangent spaces, differentiable maps, and differentials
• Submanifolds, immersions, and imbeddings
• Vector fields
• Lie groups and Lie Algebras
• Differential forms
• Integration; Stokes' Theorem

Education

The course is part of the following programs:

• MSc Programme in Mathematics
• MSc Programme in Statistics
• MSc Programme in Mathematics with a minor subject

Learning Outcome

Upon completing the course, students will have acquired the following knowledge, skills, and competences:

• Knowledge: Central definitions and theorems from the theory
• Skill:
  • Decide whether a given subset of R^n is a manifold
  • Determine the differential of a smooth map
  • Work with tangent vectors, including the Lie algebra of a Lie group
  • Utilize topological concepts in relation with manifolds
  • Find the Lie bracket of given vector fields
  • Work with exterior differentiation and pull-back of differential forms
• Competences:
  • Perform logical reasoning within the subject of the course
  • Give an oral presentation of a specific topic within the theory as well as a strategy for solving a specific problem

Teaching and Learning Methods

The course consists of 35 hours of lectures and 28 hours of exercises during 7 weeks.

Recommended Prerequisites

The recommended prerequisites for the course are:

• Analyse 1 (An1)
• Geometri 1 (Geom1)
• Topologi (Top)
• Advanced Vector Spaces (AdVec) or similar
• Academic qualifications equivalent to a BSc degree is recommended

Exam

The exam is an oral examination, 30 minutes (30-minute preparation time), and a mandatory assignment must be approved before the exam.

ECTS

The course is worth 7.5 ECTS points.

Type of Assessment

The type of assessment is oral examination.

Examination Prerequisites

A mandatory assignment must be approved before the exam.

Aid

All aids are allowed during preparation, but no aids are allowed during the examination.

Marking Scale

The marking scale is a 7-point grading scale.

Censorship Form

The censorship form is external censorship.

Re-exam

The re-exam is the same as the ordinary exam. If the assignment was not approved before the ordinary exam, the assignment must be handed in and approved three weeks before the re-exam.

Criteria for Exam Assessment

The student should convincingly and accurately demonstrate the knowledge, skills, and competences described under Intended learning outcome.

Course Type

The course type is a single subject course (day).

Workload

The workload for the course is:

• Category: Lectures
• Hours: 35
• Preparation: 142
• Theory exercises: 28
• Exam: 1
• English: 206

Language

The language of the course is English.

Course Number

The course number is NMAA06062U.

Programme Level

The programme level is Full Degree Master.

Duration

The duration of the course is 1 block.

Placement

The placement of the course is Block 2.

Schedule Group

The schedule group is B.

Capacity

There is no limitation on the number of students, unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.

Study Board

The study board is the Study Board of Mathematics and Computer Science.

Contracting Department

The contracting department is the Department of Mathematical Sciences.

Contracting Faculty

The contracting faculty is the Faculty of Science.

Course Coordinator

The course coordinator is Damian Longin Osajda.