The differential geometric analysis of higher dimensional manifolds began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized to the notion of a manifold along with other accompanying new notions such as tensors, forms, connections, Lie-derivatives etc. The motivation and momentum for this development is mainly found in the need for a precise geometric framework in which we can effectively analyse the local and global shapes of - as well as the short time and long time processes in - higher dimensional configuration spaces.
This course will cover a bouquet of results, examples, and modern applications concerning this generalized global geometric analysis.
The course aims to give the students a firm background for acquiring and applying new literature and new results from the fields of manifold analysis and advanced differential geometry.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
Recognize a topological paracompact Hausdorff space
Construct simple atlases for simple but nontrivial manifolds
Find the rank of a given diffeomorphism and apply Sard's theorem
Prove if a given subset is a submanifold
Calculate Lie derivatives of tensor fields
Discuss and explain the definition of a Riemannian manifold
Discuss, explain, and apply the basic properties of a connection
Calculate and apply the gradient, the Hessian, and the Laplacian of smooth functions on Riemannian manifolds
Suggest manifold models of simple configuration spaces
Read and acquire modern literature concerning analysis on manifolds
Master the basic theory well enough to carry out a self study of a theoretical extension or a technical application
Recognize and speculate on other advances of the elementary theory of curves and surfaces than those covered in the course
Kursusindhold:
Fundamental concepts and results which will be covered in the course include: Differentiable manifolds (recap of topology, the notion of an atlas); Tangent spaces (and tangent bundles); Smooth maps and tangent maps (diffeomorphisms); Rank of a smooth map and level sets; Vector fields (integral curves and Lie derivatives); Tensors and multilinear algebra; Tensor fields and derivations; Metric tensors (Riemannian manifolds); Differential forms; Gradient, Hessian, Laplacian, and volume forms on Riemannian manifolds.
Mulighed for GRØN DYST deltagelse:
Kontakt underviseren for information om hvorvidt dette kursus giver den studerende mulighed for at lave eller forberede et projekt som kan deltage i DTUs studenterkonference om bæredygtighed, klimateknologi og miljø (GRØN DYST). Se mere på http://www.groendyst.dtu.dk/kursustilmelding.aspx
