04/07/03
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We study the image of the bounded cohomology groups in the singular cohomology groups under the natural homomorphism (the images are called the bounded parts). These bounded parts vanish when the fundamental group is amenable. In general, however, they depend on the fundamental group as well as the high dimensional topology of the manifold. We show that for closed Riemannian n-manifold with Ricci curvature bounded below by -(n-1) and diameter bounded above by D, the dimensions of the bounded parts are bounded by a constant C(n, D). We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups among closed Riemannian n-manifolds whose sectional curvature is bounded below by -1 and diameter is bounded above by D.
We show that every open manifold admits a complete Riemannian metric of quadratic curvature decay. Then we show that there are topological obstructions for an open manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of the Euclidean space of the same dimension. Clip here for the introduction and main results.
In this paper we prove that a complete noncompact manifold with nonnegative Ricci curvature has a trivial codimension one homology unless it is a split or flat normal bundle over a compact totally geodesic submanifold. In particular, we prove the conjecture that a complete noncompact manifold with positive Ricci curvature has a trivial codimension one integer homology. We also have a corollary stating when the codimension two integer homology of such a manifold is torsion free.
On Projectively Related Einstein Metrics in Riemann-Finsler geometry. Ps Math. Ann. 320 (2001), 625-647.
In this paper we study pointwise projectively related Einstein metrics (having the same geodesics as point sets). We show that pointwise projectively related Einstein metrics are related by some simple formulas along geodesics. In particular, we show that if two pointwise projectively related Einstein metrics are complete with negative Einstein constants , then one is a multiple of another. Metrics under our consideration are not necessarily Riemannian. They are in general Finslerian Einstein metrics (Note: The notion of Ricci curvature is well defined for Finsler metrics). Article in PDF format
Conjugate Radius and Positive Scalar Curvature Ps Dvi Math. Z. 238(2001), 431-439.
In this paper, we investigate the conjugate radius of a complete open metric space with uniform positive scalar curvature. The class of metric spaces under our consideration contains Riemannian spaces with uniform positive scalar curvature.
We construct a family of Finsler metrics on the 3-sphere with K=1. They are all not locally projectively flat.
We show that if two Riemannian metrics g* and g are pointwise projectively equivalent, i.e., they have common geodesics as point sets, and the Ricci curvature of g* is less than or equal to that of g, then the projective equivalence is trivial provided that g is complete. In this case, g* is parallel with respect to g and the Riemann curvatures are equal. The curvature condition can be weakened when the manifold is compact. We actually prove this rigidity theorem for more general geometric structures, such as Finsler metrics and sprays. (March, 2001)
In this paper, we construct a non-projectively flat Randers metric with K=0 in each dimension. This Randers metric is not positively complete. We show that every positively complete Randers metric with K=0 must be locally Minkowskian.
We classify locally projectively flat Randers metrics with constant curvature and obtain a new family of Randers metrics of negative constant curvature.
The well-known Funk metric on the unit ball is a Randers metric with many special properties. In this paper, we study Randers metrics with similar properties as the Funk metric. We show that for a Randers metric of constant curvature, the mean Landsberg curvature is proportional to the mean Cartan torsion if and only if it is locally projectively flat. Then we classify all Randers metrics of constant curvature with this property.
We construct infinitely many two-dimensional Finsler metrics on the 2-sphere and 2-disk with non-zero constant Gauss curvature. They are all not locally projectively flat.
It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on a bounded convex domain in R^n such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). Thus it is a natural problem to study those of constant curvature.
In this paper, we study a special class of solutions to the Hilbert Fourth Problem in the smooth case. We first give a formula for x-analytic projective Finsler metrics F(x,y) of constant curvature K=c using a power series with coefficients expressed in terms of f(y):=F(0, y), h(y):=(1/2)F(x,y)^{-1}F_{x^k}(0, y)y^k and c. Then, for any given pair {f(y), h(y)} and any given constant c, we give an algebraic formula for smooth projective Finsler metrics with constant curvature K=c and F(0, y)=f(y) and F_{x^k}(0, y)y^k=2f(y)h(y). By these formulas, we obtain several special projective Finsler metrics of constant curvature which can be used as models in Finsler geometry.
We study compact negatively curved Finsler manifolds of dimension greater than two. We show that for any such a Finsler manifold (M, F), if the flag curvature K is independent of flags P for any fixed flagpole y, then F= a + b is a Randers metric. Further, we show that for a compact negatively curved Finsler manifold (M, F), if F is locally projectively flat, then F = a + b , where a is a Riemannian metric of constant curvature and b is a closed 1-form.
In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, we classify locally projectively flat Randers metrics with isotropic S-curvature.
The flag curvature of a Finsler metric is a
natural extension of the sectional curvature in Riemannian geometry, while the
S-curvature is a non-Riemannian quantity. Berwald metrics (including all
Riemannian metrics) form an important class of Finsler metrics. For Berwald
metrics, the S-curvature always vanishes. There are many non-Berwaldian
Finsler manifolds with constant S-curvature, some of them are of scalar
curvature or constant flag curvature. In this paper, we prove a global
rigidity theorem that every negatively curved closed Finsler manifold must be
Riemannian if its S-curvature is constant. We also study the nonpositive flag
curvature case and give some non-trivial examples.
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Open Problems (posted on 04/07/2003)
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