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 symplectic manifold, be a nonsingular symplectic vector field in the sense that . Then is a primitive of , and hence is an exact symplectic manifold.