## 20410411 - GE310 - ELEMENTS OF ADVANCED GEOMETRY

Topology: topological classification of curves and surfaces. Differential geometry: study of the geometry of curves and surfaces in R^3 to provide concrete and easily calculable examples on the concept of curvature in geometry. The methods used place the geometry in relation to calculus of several variables, linear algebra and topology, providing the student with a broad view of some aspects of mathematics.

Curriculum

teacher profile | teaching materials

Programme

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Core Documentation

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Type of delivery of the course

Type of evaluation

written and oral exam

teacher profile | teaching materials

Programme

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.

2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.

3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation

and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.

4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.

5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.

6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.

7. Homeworks.

8. 12 hours of lab for the visualization and computation on curves and surfaces.

Core Documentation

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). http://dx.doi.org/10.1007/b98853

[2] E. Sernesi, Geometria 2. Boringhieri, (1994).

[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).

[4] M. Abate, F. Tovena, Curves and Surfaces. Springer, (2006).

Type of delivery of the course

In-class lectures.

Attendance

Students are advised to attend classes regularly and to keep up to date with class content and exams.

Type of evaluation

Written and oral exam.

teacher profile | teaching materials

Mutuazione: 20410411 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 PONTECORVO MASSIMILIANO, SCHAFFLER LUCA

Programme

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Core Documentation

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Type of delivery of the course

Type of evaluation

written and oral exam

teacher profile | teaching materials

Mutuazione: 20410411 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 PONTECORVO MASSIMILIANO, SCHAFFLER LUCA

Programme

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.

2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.

3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation

and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.

4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.

5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.

6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.

7. Homeworks.

8. 12 hours of lab for the visualization and computation on curves and surfaces.

Core Documentation

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). http://dx.doi.org/10.1007/b98853

[2] E. Sernesi, Geometria 2. Boringhieri, (1994).

[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).

[4] M. Abate, F. Tovena, Curves and Surfaces. Springer, (2006).

Type of delivery of the course

In-class lectures.

Attendance

Students are advised to attend classes regularly and to keep up to date with class content and exams.

Type of evaluation

Written and oral exam.