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MATH4171: RIEMANNIAN GEOMETRY IV

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Type Open
Level 4
Credits 20
Availability Available in 2024/2025
Module Cap
Location Durham
Department Mathematical Sciences

Prerequisites

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

Corequisites

  • None.

Excluded Combinations of Modules

  • None.

Aims

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

Content

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

Learning Outcomes

Subject-specific Knowledge:

  • Have a knowledge and understanding of Riemannian geometrydemonstrated 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 argumentsand 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

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week for 20 weeks and 2 in term 3.1 Hour42 
Problems Classes8Four in each of terms 1 and 21 Hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
Written examination3 hours100 

Formative Assessment

Eight assignments to be submitted.

More information

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