10. Let , be a strictly increasing family of homeomorphisms on the unit circle, be the rotation number of . Poincare observed that if and only if admits some periodic points of period . In this case admits fixed points.

Note that is continuous, and non-decreasing. However, may not be strictly increasing. In fact, if and , then is locked at for a closed interval . More precisely, if for some , then on for some ; if ; while if both happen.

Also oberve that if , then is a singelton. So assuming is not unipotent for each , the function is a Devil’s staircase: it is constant on closed intervals , whose union is dense in .

9. Let be a vector field on , be the flow induced by on . That is, . Then we take a curve , and consider the solutions . There are two ways to take derivative:

(1) .

(2) , which induces the tangent flow of .

Combine these two derivatives together:

This gives rise to an equation

Formally, one can consider the differential equation along a solution :

, . Then is called the linear Poincare map along . Suppose . Then determines if the periodic orbit is hyperbolic or elliptic. Note that the path , contains more information than the above characterization.

8. Frink’s proof a metricization theorem. Suppose a topological space is endowed with a function satisfying (a) iff ; (b) ; (c) .

Then there is a metric on induced by :

It is easy to see that defines a metric on . Moreover, it is proved that .

More generally, the third condition can be replaced by (d) there exists a positive function such that if and , then . Define a sequence inductively by , , . Clearly . Then we discretize the function : let if . Then we check that the new function satisfies (c) and induces a metric . Clearly induces the same topology on as .

7. Given a embedding of a convex sphere , let be the Gauss map, that is, for each point , is the unit outer normal vector of at . Clearly is a diffeomorphism. The Gauss curvature can be viewed as a function on the standard through , and satisfies

. (*)

Minkowski problem. Given a smooth positive function on satisfying (*).

Is there a Riemannian metric on such that ?

This 2D version was also answered in the same paper of Nirenberg in 1953. For the high dimensional case see Pogorelov in 1969 and Cheng–Yau in 1976.

6. Weyl problem. Let be a Riemannian metric on the 2-sphere with positive Gauss curvature. Does there exist an isometric embedding of into ?

H. Lewy proved in 1938 the existence under the assumption that the metric is analytic, using his results on analytic Monge-Ampere equations. Nirenberg proved the existence in 1953 using his results on strong apriori estimates for nonlinear elliptic PDE in 2D. Aleksandrov obtained a generalized solution in 1948 as a limit of polyhedra, and Pogorelov proved the regularity of this generalized solution. In 1969 Pogorelov posed and solved Weyl’s problem for embedding into a three-dimensional Riemannian manifold.

5. Let be a -dimensional Riemannian manifold, be the sectional curvature tensor induced by the Levi–Civita connection. Clearly is determined by ? To what extend does determine ? See here.

Answered here by Misha Kapovich:

(1) if and has nowhere constant sectional curvature, then a diffeomorphism preserving sectional curvature is necessarily an isometry.

Explain: specifying the sectional curvature of a metric is generally a very overdetermined problem in higher dimensions.

(2) if , then there are counter-examples. Weinstein’s argument shows that every Riemannian surface admits different metrics of the same Gauss curvature (using flow orthogonal to the gradient of the curvature function).

4. Let be a diffeomorphism on a manifold , be an -invariant expanding foliation: there exists such that for any . Then the leaf volume grows polynomially.

Let . Then for any , pick . Or equally, . Then ,

3. Hilbert's fourth problem is to find all geometries whose axioms are closest to those of Euclidean geometry for which lines are geodesics. He also provided an example, a Finsler metric on a convex body. More precisely, let be a bounded convex body. Let . Then for each vector , draw the line through in the direction of . This line intersects at two points, say (forward and backward). Then the Finsler is given by . Clearly this definition depends on the shape of and the direction of . So it may not be Riemannian. Note that the Finsler is reversible. Moreover, the geodesic distance between two points is given by , where are the points of intersections of the line from to , .

2. Suspension. Let be a group action on a manifold . Then we consider the induced action of on the product manifold , where . Consider the quotient space , on which there is a natural action that comes from shift . Clearly the later commutes with the action and desends to an action on .

1. Let be a flow on , be the time-1 map, be the set of flow-invariant probability measures, be the set of -invariant probability measures. Clearly . On the other hand, for each , the measure is flow invariant. Therefore,

.