### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title DIFFERENTIAL GEOMETRY Code MATH349 Coordinator Dr A Rizzardo Mathematical Sciences Alice.Rizzardo@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 6 FHEQ First Semester 15

### Aims

This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space.  While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

### Learning Outcomes

(LO1) 1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces.

(LO2) 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

(LO3) 1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.

(LO4) 1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

(LO5) 2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.

(LO6) 2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.

(LO7) 3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.

(LO8) 3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.

(LO9) 4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,

(LO10) 4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.

(S1) Problem solving skills

(S2) Numeracy

### Syllabus

1. Curves in the plane and in space.
2. Surface patches in 3-space. Parametrizations.
3. Distance and the first fundamental form of a surface.
4. Curvature of surfaces and the second fundamental form. Special curves on a surface: principal curves, asymptotic curves, geodesics.  Elliptic, hyperbolic and parabolic points.
5. Gauss's theorem on curvature: the intrinsic nature of the Gauss curvature.
6. Geodesics on a surface.
7. The Gauss-Bonnet theorem.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101 Calculus I 2021-22; MATH102 CALCULUS II 2021-22; MATH103 Introduction to Linear Algebra 2021-22

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment This is an anonymous assessment.  120    70
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Class Test on campus  60    30