Affine Decomposition of Isometries in Nilpotent Lie Groups
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
Affine Decomposition of Isometries in Nilpotent Lie Groups
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
two classes of locally compact sober spaces
... considerable interest in the operator algebra community. Due to the fact that authors in theoretical computer science dealing with the denotational semantics of programming languages have been interested in T0 spaces and continuous lattices since Scott’s article [5], there is considerable interest i ...
... considerable interest in the operator algebra community. Due to the fact that authors in theoretical computer science dealing with the denotational semantics of programming languages have been interested in T0 spaces and continuous lattices since Scott’s article [5], there is considerable interest i ...
Shortest paths and geodesics
... Example 2.2.2 (Length space). The Euclidean space E n is a length space. As seen in Proposition 2.2.1 the shortest path in dI (x, y) is a straight line between x and y and in this case the Euclidean metric d(x, y) coincides with dI (x, y). Example 2.2.3 (Length space). The sphere S1 ⊂ R2 with the Eu ...
... Example 2.2.2 (Length space). The Euclidean space E n is a length space. As seen in Proposition 2.2.1 the shortest path in dI (x, y) is a straight line between x and y and in this case the Euclidean metric d(x, y) coincides with dI (x, y). Example 2.2.3 (Length space). The sphere S1 ⊂ R2 with the Eu ...
![arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact](http://s1.studyres.com/store/data/014455921_1-3cd5e27d3d632291f2fbeb8ac68a29b6-300x300.png)
arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact
... There exists a famous finite geometry of type C3 which is not covered by any building, the so-called Neumaier Geometry [42] (see also 1.17 below). It seems to be an open problem if there exist other (finite) examples of non-building Cm geometries, and if there exist geometries of type H3 (note that ...
... There exists a famous finite geometry of type C3 which is not covered by any building, the so-called Neumaier Geometry [42] (see also 1.17 below). It seems to be an open problem if there exist other (finite) examples of non-building Cm geometries, and if there exist geometries of type H3 (note that ...
Transitive actions of locally compact groups on locally contractible
... Therefore the canonical map G=PN ! G=QN is a locally trivial fiber bundle, see [35, Theorem 3.58], and hence a fibration, see [9, Theorem 4.2]. Thus there exists a map ' W X Œ0; 1 ! G=PN such that .N; '/ is contained in P . By Theorem 2.2, there exist arbitrarily small compact normal subgroups N ...
... Therefore the canonical map G=PN ! G=QN is a locally trivial fiber bundle, see [35, Theorem 3.58], and hence a fibration, see [9, Theorem 4.2]. Thus there exists a map ' W X Œ0; 1 ! G=PN such that .N; '/ is contained in P . By Theorem 2.2, there exist arbitrarily small compact normal subgroups N ...
Monoidal closed, Cartesian closed and convenient categories of
... categorical product and function space bifunctors related by an exponential law. It is well known (see e.g., [8, Theorem 3.1], [19], [23, §3] and [25, Theorem 3.3]) that with suitable restrictions on •J^ the category of α-spaces is cartesian closed. We prove below that these ^cartesian closed" expon ...
... categorical product and function space bifunctors related by an exponential law. It is well known (see e.g., [8, Theorem 3.1], [19], [23, §3] and [25, Theorem 3.3]) that with suitable restrictions on •J^ the category of α-spaces is cartesian closed. We prove below that these ^cartesian closed" expon ...
Nonparametric Inference on Shape Spaces
... diagnostics. Such a set is called a k-ad or configuration of k points. In general, each observation x = (x1 , . . . , xk ) consists of k > m points in m-dimensions (not all same). Depending on the application, the shape of a k-ad is its orbit under a group of transformations and the corresponding sh ...
... diagnostics. Such a set is called a k-ad or configuration of k points. In general, each observation x = (x1 , . . . , xk ) consists of k > m points in m-dimensions (not all same). Depending on the application, the shape of a k-ad is its orbit under a group of transformations and the corresponding sh ...
preprint
... below. Section k also contains the proof that the spaces in (k) are homeomorphic (respectively homotopy equivalent) to spaces appearing previously. To the best of our knowledge, the homeomorphisms between the spaces in (2), (4), (5) are new, even though it was well known that the spaces are homotopy ...
... below. Section k also contains the proof that the spaces in (k) are homeomorphic (respectively homotopy equivalent) to spaces appearing previously. To the best of our knowledge, the homeomorphisms between the spaces in (2), (4), (5) are new, even though it was well known that the spaces are homotopy ...
MA651 Topology. Lecture 9. Compactness 2.
... is relaxed to the reduction of just countable open covers, then we have a weaker condition called countable compactness. ...
... is relaxed to the reduction of just countable open covers, then we have a weaker condition called countable compactness. ...
Lecture 1
... for all x ∈ X and y ∈ Y . In a Galois connection (f , g ), the map f is said to be the left adjoint map, and g is said to be the right adjoint map. Dual Galois connection Show that if (f , g ) is a Galois connection from (X , 6) to (Y , 6), then (g , f ) is a Galois connection from (Y , >) to (X , > ...
... for all x ∈ X and y ∈ Y . In a Galois connection (f , g ), the map f is said to be the left adjoint map, and g is said to be the right adjoint map. Dual Galois connection Show that if (f , g ) is a Galois connection from (X , 6) to (Y , 6), then (g , f ) is a Galois connection from (Y , >) to (X , > ...
COMPACTIFICATIONS WITH DISCRETE REMAINDERS all
... (B) and (C) are equivalent when aX - X is locally compact by Theorem 7.2 of XI, [5]. That (C) and (D) are equivalent is obvious, as is (A) implies (E). Now assume (E) and let K be a compact set of countable character containing R(X). Let {Gn\n £ N] be a countable open neighborhood base for K and tak ...
... (B) and (C) are equivalent when aX - X is locally compact by Theorem 7.2 of XI, [5]. That (C) and (D) are equivalent is obvious, as is (A) implies (E). Now assume (E) and let K be a compact set of countable character containing R(X). Let {Gn\n £ N] be a countable open neighborhood base for K and tak ...
THE FARY-MILNOR THEOREM IN HADAMARD MANIFOLDS 1
... angle comparisons, its edge BC in S can be no longer than the X-distance between B and C, and hence is a geodesic in X. Thus S is isometric in the metric of X to a planar convex set, and the theorem holds for polygons. The general theorem follows by an easy limit argument. As an immediate corollary, ...
... angle comparisons, its edge BC in S can be no longer than the X-distance between B and C, and hence is a geodesic in X. Thus S is isometric in the metric of X to a planar convex set, and the theorem holds for polygons. The general theorem follows by an easy limit argument. As an immediate corollary, ...
ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION
... We recall that a topological space X is spectral whenever X is sober and the compact and open subsets are closed under finite intersections and form a base for the topology of X. Note that in particular every spectral space is compact. A continuous map f : X → Y between spectral spaces is called spe ...
... We recall that a topological space X is spectral whenever X is sober and the compact and open subsets are closed under finite intersections and form a base for the topology of X. Note that in particular every spectral space is compact. A continuous map f : X → Y between spectral spaces is called spe ...
10/3 handout
... Recall that a subset A of (X, τ ) is connected if there are no open sets U , V which disconnect A (i.e. U ∩ V ∩ A = φ, U ∩ A 6= φ, V ∩ A 6= φ, and A ⊆ U ∪ V ). Connectedness is a topological property which is not hereditary, but is preserved under continuous maps, finite products (see Thm 23.6, Munk ...
... Recall that a subset A of (X, τ ) is connected if there are no open sets U , V which disconnect A (i.e. U ∩ V ∩ A = φ, U ∩ A 6= φ, V ∩ A 6= φ, and A ⊆ U ∪ V ). Connectedness is a topological property which is not hereditary, but is preserved under continuous maps, finite products (see Thm 23.6, Munk ...
Boundaries of CAT(0) Groups and Spaces
... • X is the universal cover of a compact negatively curved Riemannian manifold and G is its fundamental group. • G = F2 the free group of rank 2, X = T4 the infinite tree of constant valence 4. The δ-hyperbolicity condition is not preserved by the direct product. Indeed if G, G0 are infinite δ-hyperb ...
... • X is the universal cover of a compact negatively curved Riemannian manifold and G is its fundamental group. • G = F2 the free group of rank 2, X = T4 the infinite tree of constant valence 4. The δ-hyperbolicity condition is not preserved by the direct product. Indeed if G, G0 are infinite δ-hyperb ...
FROM INFINITESIMAL HARMONIC TRANSFORMATIONS TO RICCI
... R e m a r k 4.1. The last result has been known (see [24]) in the case of a compact Einstein n-dimensional (n > 2) manifold (M, g) with constant scalar curvature s. In view of Example 2.2 we can formulate the following corollary. Corollary 4.1. On a compact Riemannian manifold (M, g) of dimension 2 ...
... R e m a r k 4.1. The last result has been known (see [24]) in the case of a compact Einstein n-dimensional (n > 2) manifold (M, g) with constant scalar curvature s. In view of Example 2.2 we can formulate the following corollary. Corollary 4.1. On a compact Riemannian manifold (M, g) of dimension 2 ...
On a class of transformation groups
... X which is continuousrelative to 5. Given B C 6 we will show that f-1(B) is open in Z whichwill prove (2). By definitionof 6 we can choose e C 7 such that B is aln arc componentof V. Given pE f-I(B) let TVbe Then f(W) is arewise connected, the arc componentof p in f-1(0). includedin 0, anldmeets B a ...
... X which is continuousrelative to 5. Given B C 6 we will show that f-1(B) is open in Z whichwill prove (2). By definitionof 6 we can choose e C 7 such that B is aln arc componentof V. Given pE f-I(B) let TVbe Then f(W) is arewise connected, the arc componentof p in f-1(0). includedin 0, anldmeets B a ...
Convex Sets and Convex Functions on Complete Manifolds
... The aim of this paper is to show how convex sets and functions give strong restrictions to the topology of a certain class of complete Riemannian manifolds without boundary. The idea of convexity plays an essential role for the proofs of "finiteness theorems", which give a priori estimates for the n ...
... The aim of this paper is to show how convex sets and functions give strong restrictions to the topology of a certain class of complete Riemannian manifolds without boundary. The idea of convexity plays an essential role for the proofs of "finiteness theorems", which give a priori estimates for the n ...
Topology Proceedings - topo.auburn.edu
... a compact ordered space, this notation and terminology will always be used with respect to the order; in a stably compact space, it will always be used with respect to the specialization order. The specialization order on a space X is denoted ~x. A binary relation on a space is closed if it is a clo ...
... a compact ordered space, this notation and terminology will always be used with respect to the order; in a stably compact space, it will always be used with respect to the specialization order. The specialization order on a space X is denoted ~x. A binary relation on a space is closed if it is a clo ...
ISOMETRIES BETWEEN OPEN SETS OF CARNOT GROUPS AND
... a Lie group and acts smoothly and by smooth maps, the local isometries are still not completely understood. Here, with the term ‘local isometry’ we mean isometry between open subsets. In this paper we give a complete description of the space of local isometries for those homogeneous spaces that also ...
... a Lie group and acts smoothly and by smooth maps, the local isometries are still not completely understood. Here, with the term ‘local isometry’ we mean isometry between open subsets. In this paper we give a complete description of the space of local isometries for those homogeneous spaces that also ...
§1: FROM METRIC SPACES TO TOPOLOGICAL SPACES We
... d to Y × Y endows Y with the structure of a metric space. In particular, in high school geometry, when they said that two subsets Y1 , Y2 of Rd were congruent, what was actually meant was that they are isometric as submetric spaces of (Rd , d2 ). In general, the study of properties of metric spaces ...
... d to Y × Y endows Y with the structure of a metric space. In particular, in high school geometry, when they said that two subsets Y1 , Y2 of Rd were congruent, what was actually meant was that they are isometric as submetric spaces of (Rd , d2 ). In general, the study of properties of metric spaces ...
minimal sequential hausdorff spaces
... certain graph conditions. These characterizations include parallels of those of minimal Hausdorff spaces by Bourbaki [2] in terms of open filter bases, and of those by Herrington and Long [6] in terms of arbitrary filter bases. These characterizations reveal that a number of spaces which have been the ...
... certain graph conditions. These characterizations include parallels of those of minimal Hausdorff spaces by Bourbaki [2] in terms of open filter bases, and of those by Herrington and Long [6] in terms of arbitrary filter bases. These characterizations reveal that a number of spaces which have been the ...
5. Lecture. Compact Spaces.
... 5.3. Theorem. Let a, b ∈ R be real numbers with a < b. Then the closed interval [a, b] is compact. Proof. Assume the converse. Then there is an open cover O of [a, b] which has no finite subcover. We say a closed interval J in [a, b] is bad if no finite subcollection of O covers J. In this terminolo ...
... 5.3. Theorem. Let a, b ∈ R be real numbers with a < b. Then the closed interval [a, b] is compact. Proof. Assume the converse. Then there is an open cover O of [a, b] which has no finite subcover. We say a closed interval J in [a, b] is bad if no finite subcollection of O covers J. In this terminolo ...
Non-Euclidean Geometries
... which, through a point not on a line, there are no parallels to the given line? • Saccheri already found contradiction, but based on fact that straight lines were infinite • Riemann deduced that “extended continuously” did not mean “infinitely long” ...
... which, through a point not on a line, there are no parallels to the given line? • Saccheri already found contradiction, but based on fact that straight lines were infinite • Riemann deduced that “extended continuously” did not mean “infinitely long” ...