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
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.
[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).
Mutuazione: 20410411 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 R MASCARENHAS MELO ANA MARGARIDA, SUPINO PAOLA
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.
Prerequisiti
linear algebra, vector calculus, general topologyCore Documentation
Testi consigliati[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).
Attendance
In presence: non mandatory but strongly recommendedType of evaluation
Written and oral examination teacher profile teaching materials
[2] J.W. Milnor, Topology from the differential viewpoint. Princeton University Press, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
Mutuazione: 20410411 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 R MASCARENHAS MELO ANA MARGARIDA, SUPINO PAOLA
Programme
Exercises in parallel with the lessons on the following topics: 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.Prerequisiti
Basic algebra (groups and rings), basic linear algebra (linear systems, matrices and determinants, vector spaces and linear applications, affine geometry), Euclidean geometry, especially in the plane and in space, basic general topologyCore Documentation
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). (http://dx.doi.org/10.1007/b98853)[2] J.W. Milnor, Topology from the differential viewpoint. Princeton University Press, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
Attendance
Highly recommended, lectures with exercises and active student participation three times a weekType of evaluation
Solve topology and differential geometrywritten exercises. Present a theorem with proof. Argue and prove. teacher profile teaching materials
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.
[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).
Mutuazione: 20410411 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 R MASCARENHAS MELO ANA MARGARIDA, SUPINO PAOLA
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.
Prerequisiti
linear algebra, vector calculus, general topologyCore Documentation
Testi consigliati[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).
Attendance
In presence: non mandatory but strongly recommendedType of evaluation
Written and oral examination teacher profile teaching materials
[2] J.W. Milnor, Topology from the differential viewpoint. Princeton University Press, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
Mutuazione: 20410411 GE310 - ISTITUZIONI DI GEOMETRIA SUPERIORE in Matematica L-35 R MASCARENHAS MELO ANA MARGARIDA, SUPINO PAOLA
Programme
Exercises in parallel with the lessons on the following topics: 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.Prerequisiti
Basic algebra (groups and rings), basic linear algebra (linear systems, matrices and determinants, vector spaces and linear applications, affine geometry), Euclidean geometry, especially in the plane and in space, basic general topologyCore Documentation
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). (http://dx.doi.org/10.1007/b98853)[2] J.W. Milnor, Topology from the differential viewpoint. Princeton University Press, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
Attendance
Highly recommended, lectures with exercises and active student participation three times a weekType of evaluation
Solve topology and differential geometrywritten exercises. Present a theorem with proof. Argue and prove.