100 research outputs found
Positive Quaternionic Kaehler manifolds and symmetry rank: II
Let M be a positive quaternionic Kaehler manifold of dimension 4m. If the
isometry group Isom(M) has rank at least m/2 +3, then M is isometric to HP^m or
Gr_2(C^{m+2}). The lower bound for the rank is optimal if m is even.Comment: 10 page
Finite isometry groups of 4-manifolds with positive sectional curvature
Let M be an oriented compact positively curved 4-manifold. Let G be a finite
subgroup of the isometry group of . Among others, we prove that there is a
universal constant C (cf. Corollary 4.3 for the approximate value of C), such
that if the order of G is odd and at least C, then G is either abelian of rank
at most 2, or non-abelian and isomorphic to a subgroup of PU(3) with a
presentation \{A, B| A^m=B^n=1, BAB^{-1}=A^r, (n(r-1), m)=1, r\ne
r^3=1(\text{mod}m) \}. Moreover, M is homeomorphic to CP^2 if G is non-abelian,
and homeomorphic to S^4 or CP^2 if G is abelian of rank 2.Comment: 17 page
Positive quaternionic Kaehler manifolds and symmetry rank
A quaternionic K\"ahler manifold M is called {\it positive} if it has
positive scalar curvature. The main purpose of this paper is to prove several
connectedness theorems for quaternionic immersions in a quaternionic K\"ahler
manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic
submanifolds in a positive quaternionic K\"ahler manifold. As applications we
prove that, among others, a 4m-dimensional positive quaternionic K\"ahler
manifold with symmetry rank at least (m-2) must be either isometric to \Bbb
HP^m or Gr_2(\Bbb C^{m+2}), if m\ge 10.Comment: 21 page
Positively curved manifolds with maximal discrete symmetry rank
Let M be a closed simply connected n-manifold of positive sectional
curvature. We determine its homeomorphism or homotopic type if M also admits an
isometric elementary p-group action of large rank. Our main results are: There
exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric
\Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that
M^{2n} is homeomorphic to a sphere or a complex projective space. (2) If
M^{2n+1} admits an isometric S^1 x \Bbb Z_p^k-action for a prime p\ge p(n),
then k\le n and ``='' implies that M is homeomorphic to a sphere. (3) For M in
(1) or (2), if n\ge 7 and k\ge [\frac{3n}4]+2, then M is homeomorphic to a
sphere or homotopic to a complex projective space.Comment: 18 page
Collapsed 5-manifolds with pinched positive sectional curvature
Let M be a closed 5-manifold of pinched curvature 0<\delta\le \text{sec}_M\le
1. We prove that M is homeomorphic to a spherical space form if M satisfies one
of the following conditions: (i) \delta =1/4 and the fundamental group is a
non-cyclic group of order at least C, a constant. (ii) The center of the
fundamental group has index at least w(\delta), a constant depending on \delta.
(iii) The ratio of the volume and the maximal injectivity radius is less than
\epsilon(\delta). (iv) The volume is less than \epsilon(\delta) and the
fundamental group \pi_1(M) has a center of index at least w, a universal
constant, and \pi_1(M) is either isomorphic to a spherical 5-space group or has
an odd order.Comment: 41 page
Convergence of Kaehler-Ricci flow with integral curvature bound
Let , , be a solution of the normalized
K\"ahler-Ricci flow on a compact K\"ahler -manifold with
and initial metric .
If there is a constant independent of such that then, for any , a
subsequence of converges to a compact orbifold with
only finite many singular points in the Gromov-Hausdorff sense,
where is a K\"ahler metric on satisfying the
K\"ahler-Ricci soliton equation, i.e. there is a smooth function such that
Ric(h)-h=\nabla\bar{\nabla}f, {\rm and}\it \nabla \nabla f=\bar{\nabla}
\bar{\nabla} f=0. $
Reflection groups in non-negative curvature
We provide an equivariant description/classification of all complete (compact
or not) non-negatively curved manifolds M together with a co-compact action by
a reflection group W, and moreover, classify such W. In particular, we show
that the building blocks consist of the classical constant curvature models and
generalized open books with non negatively curved bundle pages, and derive a
corresponding splitting theorem for the universal cover.Comment: 20 page
Homeomorphism Classification of positively curved manifolds with almost maximal symmetry rank
We show that a closed simply connected 8-manifold (9-manifold) of positive
sectional curvature on which a 3-torus (4-torus) acts isometrically is
homeomorphic to a sphere, a complex projective space or a quaternionic
projective plane (sphere). We show that a closed simply connected 2m-manifold
(m>4) of positive sectional curvature on which a (m-1)-torus acts isometrically
is homeomorphic to a complex projective space if and only if its Euler
characteristic is not 2. By a result of Wilking, these results imply a
homeomorphism classification for positively curved n-manifolds (n>7) of almost
maximal symmetry rank [\frac{n-1}2].Comment: 20 page
Secondary Brown-Kervaire Quadratic forms and -manifolds
In this paper we define a secondary Brown-Kervaire quadratic forms. Among the
applications we obtain a complete classification of (n-2)-connected
2n-dimensional framed manifolds up to homeomorphism and homotopy equivalence, .
In particular, we prove that the homotopy type of such manifolds determine
their homeomorphism type
An almost flat manifold with a cyclic or quaternionic holonomy group bounds
A long-standing conjecture of Farrell and Zdravkovska and independently
S.~T.~Yau states that every almost flat manifold is the boundary of a compact
manifold. This paper gives a simple proof of this conjecture when the holonomy
group is cyclic or quaternionic. The proof is based on the interaction between
flat bundles and involutions.Comment: 8 pages, to appear in the Journal of Differential Geometry. New
version of Lemma 2.5: A manifold bounds if there is an involution on TM whose
fixed bundle is full ran
- β¦