Department: Mathematical Sciences

MATH4171: RIEMANNIAN GEOMETRY IV

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

• Have a knowledge and understanding of Riemannian geometry demonstrated through the following topic areas:
• Riemannian manifolds;
• geodesics;
• Levi-Civita connection;
• curvature;
• global comparison results.

• Have developed advanced technical and scholastic skills in the area of the geometry of surfaces in 3-space.

• 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

Summative Assessment

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