Differential Geometry


The main goal is to introduce to the students classic results of Differential Geometry of curves and surfaces on  ℝ3  and  to give them the methods of geometric vision in order to provide them enough tools to approach modern geometric theories.

General characterization





Responsible teacher

Ana Cristina Malheiro Casimiro, João Pedro Bizarro Cabral


Weekly - 5

Total - 98

Teaching language



Linear Algebra I and II, Geometry, Calculus III and IV


Pressley, "Elementary differential geometry" , Springer Undergraduate Mathematics Series, 2010

M. P. Carmo, "Differential Geometry of curves and surfaces", Dover Publications, 2016.

O’Neil, "Elementary differential geometry ", Academic Press, Elsevier, 2006.

S. Montiel e A. Ros "Curves and surfaces" Graduate Studies in Mathematics, 69. AMS, Providence, RI; Real Sociedad Matemática Española, Madrid, 2005.

E. Abbena, A. Gray, S. Salamon, "Modern Differential Geometry of Curves and Surfaces with Mathematica" Chapman, Boca Raton, 2006

Teaching method

The theory is explained and motivated with examples for the student, with indication of which matters to study, problems to solve resolution and clarification of doubts about theory and problems.

Evaluation method


Frequency will be granted to any student who does not unjustifiably miss more than 1/3 of the practical classes taught. Students who have obtained it in previous school year or who have any of the special statutes provided by law are exempt from attendance.

The evaluation is carried out through Continuous evaluation or Exam evaluation.

Continuous evaluation

During the semester two tests will be carried out with a duration of 1 hour 30 minutes and an evaluation of the practical classes (ap). Each test is rated up to a maximum of 20 values ​​and the practical classes can be rated between  0 and 2 values.

 1st Test (t1): all students enrolled in the course may present themselves to the 1st test.

2nd Test (t2): all the students enrolled in the course that have obtained a frequency or have a special status may submit to the 2nd test.

Evaluation of the practical classes: the teacher provides students with a list of exercises weekly. They have to solve it outside of classes, deliver the resolution in the class where they will have to make an oral presentation. Each of these will be evaluated by teachers with an integer classification between 0 and 2 points. At the end of the semester, the simple arithmetic average of all classifications (ap).

The classification of continuous evaluation (AC) is obtained by the following formula:

AC =Min( (t1 + t2) / 2 +ap , 20 )

The student is approved in the course if AC is greater than or equal to 9.5 values. If AC is less than 16.5, the final grade of the course will be AC. If AC is greater than or equal to 16.5 values, the student can choose between obtaining a final grade of 16 values ​​or performing a supplementary examination.


All the students enrolled in the course that have obtained Frequency or have special status may submit to the Exam.

At the date and time scheduled for the Exam , any student enrolled in the course that has obtained Frequency or has special status and who has not obtained approval in the Continuous Evaluation can take the exam for 3 hours.

If the student performs the Exam (his or her classification is er) and

AE =Min( er + ap , 20)

is greater than or equal to 9.5, the student is approved. If AE is less than 16.5 the final grade of the course will be AE. If AE is greater than or equal to 16.5 values, the student can choose between obtaining a final grade of 16 values ​​or performing a supplementary examination.

Grade improvement

Students have the right to improve grade by enrollment within the established deadlines, at the time of the Exam. In this case, they will take the 3-hour Exam  as described in the previous paragraph. 



Only those students who carry an official identification document with a photograph (for example, Citizen''s Card, Identity Card, Passport, some versions of Student Cards) can carry out any of the tests, blank examination notebook.

Final considerations

In all that this Regulation is missing, the FCT-UNL General Regulations are valid.

Subject matter

1. Curves in space: parametrisation by arc-length, reparametrization, curvature, torsion, Frenet trihedron.
2. Surfaces in three dimensions: smooth surfaces, tangent space, smooth maps, differential, normal and orientability, level surfaces and quadrics, the first fundamental form, isometries.
3. Curvature of surfaces: second fundamental form, Weingarten linear map and Gauss map, normal and geodesics curvatures, parallel transport and covariant derivative, principal, mean and gaussian curvatures, flat surfaces, surface of constant mean curvature, Gaussian curvature of compact surfaces.
4. Geodesics: definition and properties, equations of geodesics, examples and applications.
5. Gauss''s Theorema Egregium.
6. The Gauss-Bonnet theorem. (optional)


Programs where the course is taught: