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# Differential Geometry - MAT00102H

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• Department: Mathematics
• Module co-ordinator: Dr. Chris Wood
• Credit value: 20 credits
• Credit level: H
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

This module will develop the classical differential geometry of curves and surfaces.

## Related modules

• None

### Prohibited combinations

• None

This is a first course in the differential geometry of curves and surfaces which requires only a background in vector calculus and some abstract linear algebra.

## Module will run

Occurrence Teaching period
A Semester 2 2024-25

## Module aims

This module will develop the classical differential geometry of curves and surfaces.

## Module learning outcomes

By the end of the module, students will be able to:

1. Compute the curvature and torsion of a space curve, and use them to classify curves geometrically.

2. Formulate the definition of a smooth surface, and differentiate smooth functions defined on them.

3. Apply the Regular Value Theorem to construct a wide variety of examples of smooth surfaces and identify their tangent spaces.

4. Formulate the first and second fundamental forms of a surface, and compute them using local coordinates.

5. Formulate the different notions of curvature of a surface, and use them to classify the points of a surface according to their geometric type.

6. Calculate the geodesic curvature of a curve on a surface, the total curvature of a polygon on a surface, and relate the total curvature of a closed surface to its topology.

## Module content

The geometry of smooth curves. Curvature; torsion; the Frenet formulas; congruence, and the Fundamental Theorem of Space Curves.

Smooth surfaces. Charts and atlases; tangent planes; the inverse function theorem; the regular value theorem; smooth mappings and their differentials; diffeomorphisms and local diffeomorphisms.

The geometry of smooth surfaces. First fundamental form (Riemannian metric); shape operator; normal curvature and principal curvatures; Gauss and mean curvatures; second fundamental form; local isometries; the "Theorema Egregium" of Gauss; the Gauss-Bonnet theorem for compact surfaces.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Differential Geometry
3 hours 100

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Differential Geometry
3 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.