A relative Toponogov comparison theorem
We present a relative form of the Toponogov comparison theorem.
math.DG↗
arXiv subjects
Jianming Wan
Publications and source records attributed to Jianming Wan.
Harmonic 2-forms and positively curved 4-manifolds
We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a Kato type inequality, then it is definite. We also discuss some new insights for compact Riemannian 4-manifolds of positive sectional curvature.
math.DG↗
Manifolds tightly covered by two metric balls
In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.
math.DG↗
An extension of Bonnet-Myers theorem
We give a complementary generalization of the extensions of Bonnet-Myers theorem obtained by Calabi and also Cheeger-Gromov-Taylor.
math.DG↗
On the fundamental group of complete manifolds with almost Euclidean volume growth
It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.
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Harmonic maps from $\mathbb{C}^{n}$ to Kahler manifolds
In this paper, we shall prove that a harmonic map from $\mathbb{C}^{n}$ ($n\geq2$) to any Kahler manifold must be holomorphic under an assumption of energy density. It can be considered as a complex analogue of the Liouville type theorem for harmonic maps obtained by Sealey.
math.DG↗
Integral of distance function on compact Riemannian manifolds
In this paper we show that, under some curvature assumptions the integral of distance function on a compact Riemannian manifold is bounded below by the product of diameter, volume and a constant only depending on the dimension.
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A result on Ricci curvature and the second Betti number
We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction.
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Curvature and integrability of almost complex structures
This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the complex structures on an almost complex manifold. I also discuss the almost-Hermitian case of harmonic complex structures and the connections with balanced structures.
math.DG↗
Degrees of periods
We introduce the concept of degree to classify the periods in the sense of Kontsevich. Using this notion we give some new understanding of some problems in transcendental number theory.
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Harmonic Complex Structures
In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and Kähler structure. From differential geometric viewpoint, it is a natural generalization of Kähler structure.
math.DG↗