Part I: Theory of Surfaces


1. Immersions in Euclidean spaces

1.1 Regular surfaces

1.1.1 Immersions on the unit disc

1.1.2 The concept of an immersion

1.1.3 Normal space and normal bundle

1.2 Examples

1.2.1 Surface graphs

1.2.2 The complexified Neile parabola

1.2.3 The Henneberg minimal surface

1.3 Orthonormal normal frames and examples

1.3.1 Definition

1.3.2 ONFs for holomorphic surfaces

1.3.3 ONFs for surface graphs

1.3.4 ONFs for spherical surfaces

1.3.5 ONFs for the Clifford torus

1.3.6 ONFs for surfaces of Killing

1.3.7 ONFs for set products of curves

1.3.8 ONFs for generalized rotational surfaces

2. Fundamental forms

2.1 The first fundamental forms

2.1.1 Definition

2.1.2 The line element

2.2 The second and the third fundamental form

2.2.1 Definition of the second fundamental form

2.2.2 Definition of the third fundamental form

2.3 Curves on surfaces

2.3.1 The length of a curve on a surface

2.3.2 The angle between two curves on a surface

2.4 Parameter systems

2.4.1 Conformal parameters

2.4.2 Geodesic polar coordinates

2.4.3 Vector fields

2.5 Parameter transformations

2.5.1 Regular parameter transformations

2.5.2 Transformation of the first fundamental form

2.5.3 Transformation of the area element


3. Differential equations and curvatures

3.1 The Gauss equations

3.1.1 The inverse first fundamental form

3.1.2 Transformation behaviour of the inverse first fundamental form I

3.1.3 Transformation behaviour of the inverse first fundamental form II

3.1.4 The Christoffel symbols

3.1.5 Transformations behaviour of the Christoffel symbols

3.1.6 The Gauss equations

3.1.7 The Gauss equations in conformal parameters

3.2 The Weingarten equations

3.2.1 The torsion coefficients

3.2.2 Transformation behaviour of the torsion coefficients

3.2.3 The Weingarten equations

3.2.4 The Weingarten equations in conformal parameters

3.3 The mean curvature vector

3.3.1 Definition of the mean curvature vector

3.3.2 Parameter invariance of the mean curvature vector

3.3.3 ONF invariance of the mean curvature vector

3.3.4 The mean curvature system

3.4 The Gauss curvature

3.4.1 Definition of the Gauss curvature

3.4.2 Parameter invariance of the Gauss curvature

3.4.3 ONF invariance of the Gauss curvature

3.4.4 The area of the spherical image

3.6 Weingarten forms and principal curvatures

3.6.1 Weingarten forms

3.6.2 Principal curvatures


4. Integrability conditions


5. Normal curvatures


6. The Gauss-Osserman map