# Geometriae Dedicata

Geometriae Dedicata > 1997 > 64 > 1 > 85-123

*L*

_{2}(q) and...

Geometriae Dedicata > 1997 > 64 > 1 > 17-40

*H*is a hyperplane of a projective space

*P*, and the point line geometry Γ has an embedding in

*P*, then the pullback from

*H*is a geometric hyperplane of Γ. We show that all geometric hyperplanes arise in this way for polar spaces of type

*D*

_{ n }, the Grassmann...

Geometriae Dedicata > 1997 > 64 > 1 > 41-53

Geometriae Dedicata > 1997 > 64 > 1 > 55-68

*n*-gons of diameter 1 is at most

*n*2nsin (π/2n). Equality is attained here if and only if

*n*has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal

*n*-gons. In fact, the convex

*n*-gons of diameter 1 and perimeter

*n*2n sin (π/2n) are in bijective correspondence with the solutions of a diophantine problem.

Geometriae Dedicata > 1997 > 64 > 1 > 69-83

*S*be a closed orientable surface of genus at least 2 and let $$\widetilde S$$ to S be a connected finite abelian covering with covering group $G$. The lifts of liftable mapping classes of

*S*determine a central extension (by

*G*) of a subgroup of finite index of the mapping class group of

*S*. This extension acts on H

_{1}( $$\widetilde S$$ ). With a few exceptions for genus 2, we determine the...

Geometriae Dedicata > 1997 > 64 > 1 > 1-16

Geometriae Dedicata > 1997 > 64 > 2 > 229-251

*X*,

*L*) be a quasi-polarized variety, i.e.

*X*is a smooth projective variety over the complex numbers $$\mathbb{C}$$ and

*L*is a nef and big divisor on

*X*. Then we conjecture that

*g*(

*L*) =

*q*(

*X*), where

*g*(

*L*) is the sectional genus of

*L*and $$q(X) = \dim H^1 (\mathcal{O}_X )$$ . In this paper, we treat the case $$\dim X = 2$$ . First we prove that this conjecture is true for $$\kappa...

Geometriae Dedicata > 1997 > 64 > 2 > 157-191

Geometriae Dedicata > 1997 > 64 > 2 > 125-144

*M*

^{3}into $$\mathbb{R}^4 $$ is rigid. The role of the Pick form in the characteristic...

Geometriae Dedicata > 1997 > 64 > 2 > 145-155

*n*-dimensional hypercube on its

*q*-dimensional cells. This group action is interpreted as $$C_2^n \cdot S_n $$ acting on the set of unit basis vectors in

**R**

_{ n }and their opposites. A kind of generating function that yields all these polynomials at once is obtained by Möbius inversion. The same technique is...

Geometriae Dedicata > 1997 > 64 > 2 > 193-227

*C*

^{∞}embedding

*F*of a

*C*

^{∞}Riemannian

*n*-manifold (

*M*,

*g*) in

*E*

^{ n+1}which is

*short*in the sense that the metric induced by

*F*is less than

*g*, there is a

*C*

^{1}isometric embedding which is arbitrarily

*C*

^{0}-close to

*F*. We extend the Nash--Kuiper result for compact

*M*to the case of deformations. In other words, we prove that given a continuous family of short

*C*...

Geometriae Dedicata > 1997 > 64 > 3 > 277-282

*n*-simplex in this note. This answers a question of W. Fenchel raised in his book, Elementary Geometry in Hyperbolic Space, (De Gruyter, Berlin, 1989, p. 174) where he obtained some necessary conditions for which six numbers have to satisfy in order to be the dihedral angles...

Geometriae Dedicata > 1997 > 64 > 3 > 261-276

*H=G/K*where

*G*is a compact simply connected simple Lie group,

*T*a maximal torus of

*G*and

*F(T,H)*=|

*E*

_{1},...,

*E*

_{m}is the fixed point set of

*T*on

*H*, then for each pair

*E*

_{ i },

*E*

_{ j }there is a 2-dimentional sphere

*N*

_{ ij }⊂

*H*such that

*E*

_{i}and

*E*

_{j}are antipodal points of

*N*

_{ij}.

Geometriae Dedicata > 1997 > 64 > 3 > 283-295

*K*be a field of characteristic 2 and let

*V*be a vector space of dimension 2

*m*over

*K*. Let

*f*be a non-degenerate alternating bilinear form defined on

*V*×

*V*. The symplectic group Sp(2

*m*,

*K*) acts on the exterior powers Λ

^{ k }

*V*for 0 ≤

*k*.≤ 2

*m*There is a contraction map ∂ defined on the exterior algebra Λ, which commutes with the Sp(2

*m*,

*K*) action and satisfies ∂

^{2}= 0 and ∂(Λ

^{ k }

*V*) ≤ Λ

^{ k−1}

*V*We prove that...

Geometriae Dedicata > 1997 > 64 > 3 > 343-354

*K*in

**R**

^{2}with illumination body

*K*

^{δ}we consider an expression connected with the affine surface area of

*K*, namely the volume differences vol

_{2}(

*K*

^{δ}) - vol

_{2}(

*K*). We investigate what kind of functions can occur as such volume differences and obtain a result similar to the one obtained in the case of floating bodies.

Geometriae Dedicata > 1997 > 64 > 3 > 297-310

^{3}bounds a simply connected orientable 4-manifold M

^{4}.’ More precisely, an edge-coloured graph representing M

^{4}is obtained as the final result of a finite and well-determined sequence of ‘admissible moves’, starting from any given edge-coloured graph...

Geometriae Dedicata > 1997 > 64 > 3 > 373-381

*n*≥ 3 whose scalar curvature S(x) is positive for all x in

*M*. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on −Δ

*u*+[(

*n*−2)/(4(

*n*−1))]

*Su*=

*qu*

^{(n+2)/(n−2)}on (

*M*,

*g*). This study finds restrictions on the existence of an injective conformal immersion...

Geometriae Dedicata > 1997 > 64 > 3 > 319-330

*M*is defined to be the energy of the mapping

*M*→

*T*

_{1}

*M*, where the unit tangent bundle

*T*

_{1}

*M*is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional...

Geometriae Dedicata > 1997 > 64 > 3 > 253-260

*L*

^{p}-bound on the negative part of the Ricci curvature tensor, p > 2. In earlier work we proved this under the assumption that the Ricci curvature is pointwise bounded from below.

Geometriae Dedicata > 1997 > 64 > 3 > 331-341