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1.
Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C
∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.
相似文献
2.
In this paper, we characterize the dynamic of every Abelian subgroups
of
,
or
. We show that there exists a
-invariant, dense open set U in
saturated by minimal orbits with
a union of at most n
-invariant vector subspaces of
of dimension n−1 or n−2 over
. As a consequence,
has height at most n and in particular it admits a minimal set in
.
This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15 相似文献
3.
Let be the homogeneous tree with degree q + 1 ≥ 3 and a finitely generated group whose Cayley graph is . The associated lamplighter group is the wreath product , where is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric
boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this
geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary.
We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary,
if the lamplighter “operates at bounded range”.
Supported by ESF program RDSES and by Austrian Science Fund (FWF) P15577. 相似文献
4.
We study hypersurfaces in Euclidean space
whose position vector x satisfies the condition L
k
x = Ax + b, where L
k
is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed
,
is a constant matrix and
is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form
, with
. This extends a previous classification for hypersurfaces in
satisfying
, where
is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A
53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].
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5.
We prove that the properties of acting metrically properly on some space with walls or some CAT(0) cube complex are closed
by taking the wreath product with . We also give a lower bound for the (equivariant) Hilbert space compression of in terms of the (equivariant) Hilbert space compression of H.
A la mémoire de Michel Matthey 相似文献
6.
We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity
on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović
and A. Katok.
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7.
Thomas Eckl 《Geometriae Dedicata》2008,137(1):149-162
Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities
in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .
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8.
Christopher Kennedy 《Algebras and Representation Theory》2006,9(5):525-537
Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman. 相似文献
9.
We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization
for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the
quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two
directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of
distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces
of non-positive curvature.
相似文献
10.
Juan A. Aledo José M. Espinar José A. Gálvez 《Calculus of Variations and Partial Differential Equations》2007,29(3):347-363
We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature
c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or .
J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de
Castilla-La Mancha, grant no. PAI-05-034.
J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta
de Andalucía Grant No. FQM325. 相似文献
11.
Nic Koban 《Geometriae Dedicata》2007,124(1):133-141
We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F
n
. We give an example involving the direct product of the Baumslag–Solitar group BS1,2 with itself.
相似文献
12.
Naoki Murabayashi 《Mathematische Annalen》2008,342(3):657-671
It is known that in the moduli space of elliptic curves, there exist precisely nine -rational points represented by an elliptic curve with complex multiplication by the maximal order of an imaginary quadratic
field. In Murabayashi and Umegaki (J Algebra 235:267–274, 2001) and Umegaki [Determination of all -rational CM-points in the moduli spaces of polarized abelian surfaces, Analytic number theory (Beijng/Kyoto, 1999). Dev.
Math., vol 6. Kluwer, Dordrecht, pp 349–357, 2002] we determined all -rational points in (the moduli space of d-polarized abelian surfaces) represented by a d-polarized abelian surface whose endomorphism ring is isomorphic to the maximal order of a quartic CM-field by using the result
in Murabayashi (J Reine Angew Math 470:1–26, 1996). In this paper, we prove that polarized abelian surfaces corresponding
to these -rational CM points have a -rational model by constructing certain Hecke characters. 相似文献
13.
14.
Lewis Bowen 《Geometriae Dedicata》2007,124(1):37-67
The classical prime geodesic theorem (PGT) gives an asymptotic formula (as x tends to infinity) for the number of closed geodesics with length at most x on a hyperbolic manifold M. Closed geodesics correspond to conjugacy classes of π1(M) = Γ where Γ is a lattice in G = SO(n,1). The theorem can be rephrased in the following format. Let be the space of representations of into Γ modulo conjugation by Γ. is defined similarly. Let be the projection map. The PGT provides a volume form vol on such that for sequences of subsets {B
t
}, satisfying certain explicit hypotheses, |π−1(B
t
)| is asymptotic to vol(B
t
). We prove a statement having a similar format in which is replaced by a free group of finite rank under the additional hypothesis that n = 2 or 3.
相似文献
15.
John P. Steinberger 《Results in Mathematics》2008,51(3-4):319-338
If is any ring or semi-ring (e.g., ) and G is a finite abelian group, two elements a, b of the group (semi-)ring are said to form a factorization of G if ab = rΣ
g∈G
g for some . A factorization is called quasiperiodic if there is some element g ∈ G of order m > 1 such that either a or b – say b – can be written as a sum b
0 + ... + b
m−1 of m elements of such that ab
h
= g
h
ab
0 for h = 0, ... , m − 1. Hajós [5] conjectured that all factorizations are quasiperiodic when and r = 1 but Sands [15] found a counterexample for the group . Here we show however that all factorizations of abelian groups are quasiperiodic when and that all factorizations of cyclic groups or of groups of the type are quasiperiodic when . We also give some new examples of non-quasiperiodic factorizations with for the smaller groups and .
Received: May 12, 2006. Revised: October 3, 2007. 相似文献
16.
Humio Ichimura 《Archiv der Mathematik》2006,87(6):539-545
Let p be an odd prime number and
. Let
be the classical Stickelberger ideal of the group ring
. Iwasawa [6] proved that the index
equals the relative class number
of
. In [2], [4] we defined for each subgroup H of G a Stickelberger ideal
of
, and studied some of its properties. In this note, we prove that when
mod 4 and [G : H] = 2, the index
equals the quotient
.
Received: 13 January 2006 相似文献
17.
It is classically known that a real cubic surface in cannot have more than one solitary point (or -singularity, locally given by x
2 + y
2 + z
2 = 0) whereas it can have up to four nodes (or -singularity, locally given by x
2 + y
2 − z
2 = 0). We show that on any surface of degree d ≥ 3 in the maximum possible number of solitary points is strictly smaller than the maximum possible number of nodes. Conversely,
we adapt a construction of Chmutov to obtain surfaces with many solitary points by using a refined version of Brusotti’s Theorem.
Combining lower and upper bounds, we deduce: , where denotes the maximum possible number of solitary points on a real surface of degree d in . Finally, we adapt this construction to get real algebraic surfaces in with many singular points of type for all k ≥ 1.
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18.
We introduce a new existence result for compact normal geodesic graphs with constant mean curvature and boundary in a class
of warped product spaces. In particular, our result includes that of normal geodesic graphs with constant mean curvature in
hyperbolic space over a bounded domain in a totally geodesic .
相似文献
19.
Li-Ping Huang 《Geometriae Dedicata》2009,138(1):1-12
Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by the k-dimensional Grassmann space on . Let be a map. This paper proves the following are equivalent: (i) is an adjacency preserving bijection in both directions. (ii) is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is
extended to the case of Bezout domains: If is an adjacency preserving bijection in both directions, then is induced by either a collineation or the duality of a collineation.
Project 10671026 supported by National Natural Science Foundation of China. 相似文献
20.
Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S
n
, and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC. 相似文献