Differential Geometry (MAST90143)

Graduate courseworkPoints: 12.5Not available in 2025

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Overview

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This subject extends notions from calculus, linear algebra and differential equations to study spaces with geometric structures.  The concepts introduced are of great importance in mathematics, physics, and all areas in which local properties of spaces are used to model systems.  

Topics include: smooth manifolds, vector bundles, multilinear algebra; Frobenius’ theorem, exterior differentiation, Lie differentiation, flows of vector fields; connections and curvature; bilinear forms, metrics, length, volume, Levi-Civita connection; parallel transport, geodesics, holonomy; connections on principal bundles; examples including Lie groups, hyperbolic geometry and homogeneous spaces.

Additional topics may include: second fundamental form and minimal submanifolds; Jacobi fields and applications to topology; constant curvature and Einstein metrics; Hodge star operator, Hodge Laplacian and harmonic forms; Lorentzian geometry and Einstein's equations; Kähler geometry; symplectic geometry; gauge theory.


Intended learning outcomes

After completing this subject, student should be able to:

  • demonstrate understanding of the basic notions of Differential Geometry, including smooth manifolds, vector bundles, Riemannian metrics, connections and curvature;
  • functionally use connections, curvature and geodesics;
  • perform geometric calculations in local coordinates;
  • explain and apply major foundational results in differential geometry;
  • demonstrate knowledge of important examples of Lie groups and homogeneous spaces; and
  • pursue further studies in differential geometry and related areas.

Generic skills

In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

  • Problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
  • Analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
  • Collaborative skills: the ability to work in a team; and
  • Time-management skills: the ability to meet regular deadlines while balancing competing commitments.

Last updated: 4 March 2025