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Durham University

Faculty Handbook 2022-2023

Module Description

Please ensure you check the module availability box for each module outline, as not all modules will run each academic year.

Department: Mathematical Sciences

MATH4171: RIEMANNIAN GEOMETRY IV

Type Open Level 4 Credits 20 Availability Available in 2022/23 Module Cap Location Durham

Prerequisites

  • Mathematics modules to the value of 100 credits in Years 2 and 3, with at least 40 credits at Level 3 and including Differential Geometry III (MATH3021).

Corequisites

  • None.

Excluded Combination of Modules

  • None.

Aims

  • Provide a knowledge of the intrinsic geometry of Riemannian manifolds. This is a significant generalisation of the metric geometry of surfaces in 3-space.

Content

  • The metric geometry of Riemannian manifolds.
  • Geodesics.
  • Various notions of curvature, and their effect on the geometry of a Riemannian manifold.
  • Second variation formula, global comparison theorems with applications.

Learning Outcomes

Subject-specific Knowledge:
  • Have a knowledge and understanding of Riemannian geometry demonstrated through the following topic areas:
  • Riemannian manifolds;
  • geodesics;
  • Levi-Civita connection;
  • curvature;
  • global comparison results.
Subject-specific Skills:
  • Have developed advanced technical and scholastic skills in the area of the geometry of surfaces in 3-space.
Key Skills:
  • Have developed an appreciation of high-level mathematical reasoning.
  • Have developed the ability to present well-reasoned arguments and operate in complex and specialised contexts.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

  • Teaching is by lectures through which the main body of knowledge is made available.
  • Students do regular formative work solving problems to gain insight into the details of the relevant theories and procedures.
  • End of year examinations assess the learning.

Teaching Methods and Learning Hours

Activity Number Frequency Duration Total/Hours
Lectures 42 2 per week for 20 weeks and 2 in term 3. 1 Hour 42
Problems Classes 8 Four in each of terms 1 and 2 1 Hour 8
Preparation and Reading 150
Total 200

Summative Assessment

Component: Examination Component Weighting: 100%
Element Length / duration Element Weighting Resit Opportunity
Written examination 3 hours 100%

Formative Assessment:

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.


Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University



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