Titlebook: Differential Geometry; From Elastic Curves Ulrich Pinkall,Oliver Gross Textbook‘‘‘‘‘‘‘‘ 2024 The Editor(s) (if applicable) and The Author(

否决

Curves in Curves in the plane . are special in several respects: For a closed plane curve . an enclosed area . can be defined, providing another geometric functional in addition to length and bending energy.

尖叫

Parallel Normal FieldsFor curves . there is an analog .of the curvature function of a plane curve. In the context of unit speed curves, this function . determines . up to an orientation-preserving rigid motion of .. Before we can define ., we have to study . along a curve in ..

GRILL

Kindle

Surfaces and Riemannian GeometryThe most simple quantity of a one-dimensional curve . is its speed ..

Hectic

道学气

确保

Levi-Civita ConnectionThe . of a surface . provides a geometrically meaningful way to take directional derivatives of a vector field .  on .. Based on the Levi-Civita connection we derive two important equations that are satisfied by the curvature of a surface in .: the . and the ..

摆动

Calculus

Closed SurfacesWe define a . as a surface . whose boundary components have been matched in pairs in such a way that . as well as its unit normal . are continuous across the boundary. This allows us to prove an analog of the fact that the tangent winding number of a closed plane curve is an integer.

去世

Variations of SurfacesWe derive the basics of Vector Calculus on surfaces and explore variations of surfaces. In particular, we compute the variational derivative of the area form . and of the shape operator .. We show that the critical points of the area functional are the surfaces with mean curvature ..