# Chinese Annals of Mathematics, Series B

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 67-72

**Z**component of the Atiyah-Patodi-Singer η invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi...

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 53-66

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 31-52

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 95-112

*n*) act in the standard way on ℂ

^{ n }and extend this action in the usual way to ℂ

^{ n+1}= ℂ ⊕ ℂ

^{ n }. It is shown that a nonsingular special Lagrangian submanifold

*L*⊂ ℂ

^{ n+1}that is invariant under this SO(

*n*)-action intersects the fixed ℂ ⊂ ℂ

^{ n+1}in a nonsingular real-analytic arc

*A*(which may be empty). If

*n*> 2, then

*A*has no compact component. Conversely, an embedded, noncompact nonsingular...

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 73-94

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 1-30

*n*-dimensional spherical or hyperbolic geometry,

*n*≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in

*S*

^{ n }(1) and

*H*

^{ n }(—1).

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 153-168

^{7}as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere

*S*

^{6}. It is known that a cone over a surface

*M*in

*S*

^{6}is an associative submanifold of ℝ

^{7}if and only if

*M*is almost complex in

*S*

^{6}. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in

*S*

^{6}are the equation for primitive...

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 143-152

*n*-manifold (

*M, g*) with Ric (

*M*) ≥ -(

*n*- 1) and Diam (

*M*) ≤

*D*are bounded by a number depending on

*D*and

*n*. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups $$ {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{H}^{*}...

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 169-178

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 193-218

*n*-dimensional manifold without boundary, embedded in ℝ

^{ n+1}, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of

*M*in a hyperplane

*X*

_{ n+1}=constant in case

*M*satisfies: for any two points (

*X*′,

*X*

_{ n+1}), $$ {\left( {{X}\ifmmode{'}\else$'$\fi,\ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n...

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 179-192

*K*

_{ X }, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 219-242

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 121-142

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 113-120

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 287-302

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 311-328

*p*

^{ n }

*q*+

*p*

^{ m }

*q*+

*q*-4 and a nontrivial element in the stable homotopy group of Moore spectum of degree

*p*

^{ n }

*q*+

*p*

^{ m }

*q*+

*q*-3 which are represented by

*h*

_{0}(

*h*

_{ m }

*b*

_{n-1}-

*h*

_{ n }

*b*

_{ m-1}) and

*i*

_{*}(

*h*

_{0}

*h*

_{ n }

*h*

_{ m }) in the

*E*

_{2}-terms of the Adams spectral sequence respectively, where

*p*≥ 7 is...

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 353-362

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 329-340

_{2}and proves that it is represented by 2

*b*

_{2,0}

*h*

_{1,2}in the ASS. The authors also compute the higher May differential of

*b*

_{2,0}, from which it is proved that $$ \widetilde{\gamma }_{s} {\left( {b_{0} h_{n} - h_{1} b_{{n - 1}} } \right)} $$ for 2 ≤

*s*<

*p*- 1 are permanent cycles in the ASS.

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 303-310

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 341-352

*n*, the prime variable equation

*n*=

*p*

_{1}+

*p*

_{2},

*p*

_{1}∈

*P*

_{γ}is solvable, with 13=15 < γ ≤ 1, where

*P*

_{γ}= {

*p*∣ $$ p = {\left[ {m^{{\frac{1} {\gamma }}} } \right]} $$ ; for integer

*m*and prime

*p*} is the set of the Piatetski-Shapiro primes.