I wonder if this could be “the start of something new”.

Let’s start with sub-Riemannian geometry and tag my posts “sR” and “maths”. I may put up other findings as well. Apologies for links that require institutional access, such as academic journals related to my studies.

3 July 2014 20:55:50

My university’s maths blog »

Q.A1 Why \phi(0)=1?

Q.A2 What reparametrization leads to \dot{\gamma}=u_1X+u_2Y with u_1^2+u_2^2=1 a.e. and length as specified?

Q.A3 Finish Chapter 3.

Q.B1 Montgomery‘s proof of abnormal geodesics (compared with the one in the book).

Q.C1 R. Monti’s paper on the regularity problem for sub-Riemannian geodesics.

Log of activity since 4th July 2014. Rough work in my hands.

Useful information, unsorted

  • All things Lie: Given (A,*), * denoting multiplication, and a,b\in A, the Lie bracket [\cdotp,\cdotp] is defined by [a,b]=a*b-b*a. The Lie algebra \,\mathfrak{g} is a vector space over a field F  with the binary operation [\cdotp,\cdotp].
    • Question: Given two vector fields X,Y:S\subseteq \mathbb{R}^n \rightarrow\mathbb{R}^n, how does one find the Lie bracket [X,Y]?
  • contact (of a distribution) : given frame fields defined near a point q, the vectors X_i(q) and Lie brackets [X_i,X_j](q) form a basis for the tangent space at q.
  • fiber of a map q:x \rightarrow y: any subset of the form q^{-1}(y) \subseteq X for some y \in Y.
  • H^1: Heisenberg group, sub-Riemannian
  • a.e.: almost everywhere
  • reparametrization: given a parametrized curve \gamma:(a,b)\rightarrow \mathbb{R}^n, a parametrized curve \tilde{\gamma}:(c,d)\rightarrow \mathbb{R}^n is a reparametrization of \gamma if there exists a diffeomorphism (=bijective and smooth function with smooth inverse) \phi: (c,d)\rightarrow (a,b) such that \tilde{\gamma}(\tau)=\gamma(\phi(\tau)), \forall\tau\in (c,d).
Pages to go over (again):
  • SpringerLink
    • Andrew Pressley, Elementary Differential Geometry
    • John M. Lee, Introduction to Smooth Manifolds
    • John M. Lee, Introduction to Topological Manifolds
    • Stefani, G., Boscain, U., Gauthier, J.-P., Sarychev, A., Sigalotti, M. (Eds.), Geometric Control Theory and Sub-Riemannian Geometry
Target: each p.39 and beyond. What is “rank k<n” in P. 2/16?
Asked Marco, tutor, about differential forms on Riemannian geometry, 18 June 2014