Topological equivalence and rigidity of flows on certain solvmanifolds

There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

Computational anatomy aims at developing models to understand the anatomical variability of organs and tissues. A widely used and validated instrument for comparing the anatomy in medical images is non-linear diffeomorphic registration which is based on a rich mathematical background. For instance, the large deformation diffeomorphic metric mapping (LDDMM) framework defines a Riemannian setting by providing a right invariant metric on the tangent spaces, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. A simpler alternative based on stationary velocity fields (SVF) has been proposed, using the one-parameter subgroups from Lie groups theory. In spite of its better computational efficiency, the geometrical setting of the SVF is more vague, especially regarding the relationship between one-parameter subgroups and geodesics. In this work, we detail the properties of finite dimensional Lie groups that highlight the geometric foundations of one-parameter subgroups. We show that one can define a proper underlying geometric structure (an affine manifold) based on the canonical Cartan connections, for which one-parameter subgroups and their translations are geodesics. This geometric structure is perfectly compatible with all the group operations (left, right composition and inversion), contrarily to left- (or right-) invariant Riemannian metrics. Moreover, we derive closed-form expressions for the parallel transport. Then, we investigate the generalization of such properties to infinite dimensional Lie groups. We suggest that some of the theoretical objections might actually be ruled out by the practical implementation of both the LDDMM and the SVF frameworks for image registration. This leads us to a more practical study comparing the parameterization (initial velocity field) of metric and Cartan geodesics in the specific optimization context of longitudinal and inter-subject image registration.Our experimental results suggests that stationarity is a good approximation for longitudinal deformations, while metric geodesics notably differ from stationary ones for inter-subject registration, which involves much larger and non-physical deformations. Then, we turn to the practical comparison of five parallel transport techniques along one-parameter subgroups. Our results point out the fundamental role played by the numerical implementation, which may hide the theoretical differences between the different schemes. Interestingly, even if the parallel transport generally depends on the path used, an experiment comparing the Cartan parallel transport along the one-parameter subgroup and the LDDMM (metric) geodesics from inter-subject registration suggests that our parallel transport methods are not so sensitive to the path.

A Lie group G is said to be uniformly finitely generated by one-parameter subgroups exp(tXi), i = 1, . . . , n, if there exists a positive integer k such that every element of G may be expressed as a product of at most k elements chosen alternatively from these one-parameter subgroups. In this paper we construct sets of left invariant vector fields on SO(n), in particular, pairs {A, B}, whose one-parameter subgroups uniformly finitely generate SO(n) and find an upper bound on the order of generation of SO(n,R) by these subgroups. We give special attention to the case n = 3. 0. Introduction. If the Lie algebra of a connected Lie group G is generated by the elements X1, . . . , Xn, then every element of G may be expressed as a finite product of elements of the form exp(tXi), where t is real and i = 1, . . . , n (Jurdjevic and Sussmann [6]). However, the number of elements required for g ∈ G may not be uniformly bounded as g ranges through G. If, in addition, G is compact and exp(tXi), i = 1, . . . , n are also compact, then it follows from Theorem 1.1 that there exists a positive integer k such that every element of G may be expressed as a product of at most k elements from exp(tXi), i = 1, . . . , n. That is, G is uniformly finitely generated by these oneparameter subgroups with order of generation k. For two and three-dimensional Lie groups, the problem has been completely solved by Koch and Lowenthal. In [1], Crouch and the present author take the initial steps in the problem of uniform finite generation of SO(n,R) (the real n(n− 1)/2-dimensional special orthogonal group with Lie algebra so(n)) and concentrate on finding pairs of generators for so(n), orthogonal with respect to the killing form 〈·, ·〉 and whose one-parameter subgroups uniformly finitely generate SO(n). This paper is still devoted to the uniform generation problem of SO(n). Section 1 is introductory. Sections 2 and 3 are concerned with Work supported in part by Centro de Matematica da Universidade de CoimbraINIC and by JNICT under project 87.62. Received by the editors on May 22, 1984 and in revised form on May 27, 1988. Copyright c ©1991 Rocky Mountain Mathematics Consortium

Let $\mathscr{S}_{r}$, be the real topological vector space of real-valued rapidly decreasing functions and let $\mathcal{O}(\mathscr{S}_{r})$ be the group of rotations of $\mathscr{S}_{r}$. Then every one-parameter subgroup of $\mathcal{O}(\mathscr{S}_{r})$ induces a flow in $\mathscr{S}_{r}^{*}$ the conjugate space of $\mathscr{S}_{r}$ with the Gaussian White Noise as an invariant measure. The author constructed a group of functions which is isomorphic to a subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and some of its one-parameter subgroups. But the problem whether it contains sufficiently many one-parameter subgroups has been a problem. In Part I of the present paper, we answer this problem affirmatively by constructing two classes of one-parameter subgroups in a concrete way. In Part II, we construct an infinite dimensional Lie subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and the corresponding Lie algebra. Namely, we construct a topological subgroup $\mathfrak{G}$ of $\mathcal{O}(\mathscr{S}_{r})$ which is coordinated by the nuclear space $\mathscr{S}_{r}$ and the algebra $\mathfrak{a}$ of generators of one-parameter subgroups of $\mathfrak{G}$ which is closed under the commutation. Furthermore, we establish the exponential map from $\mathfrak{a}$ into $\mathfrak{G}$ and prove continuity.

UDC 515.1 We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \ddagger} /} G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G$. It was pointed out by Lipsman that the correspondence between $\hat{G} $ and ${\mathfrak{g}^{ \ddagger} /} G$ is bijective. Under some assumption on $G$, we give another proof for the continuity of the orbit mapping (Lipsman mapping)$$\Theta : {\mathfrak{g}^{ \ddagger} /} G - \rightarrow \hat{G} .$$

Let Γ \Gamma and Γ ′ \Gamma ’ be lattices, and ϕ \phi and ϕ ′ \phi ’ one-parameter subgroups of the connected Lie groups G G and G ′ G’ . If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces G / Γ G/\Gamma and G ′ / Γ ′ G’ /\Gamma ’ are topologically equivalent, then they are topologically equivalent by an affine map. (a) G G and G ′ G’ are one-connected and nilpotent. (b) G G and G ′ G’ are one-connected and solvable, and for all X X in L ( G ) L(G) and X ′ X’ in L ( G ′ ) L(G’ ) , ad ⁡ ( x ) \operatorname {ad} (x) and ad ⁡ ( X ′ ) \operatorname {ad} (X’ ) have only real eigenvalues, (c) G G and G ′ G’ are centerless and semisimple with no compact direct factor and no direct factor H H isomorphic to PSL ⁡ ( 2 , R ) \operatorname {PSL} (2,\,R) such that Γ H \Gamma H is closed in G G . Moreover, in condition (c), the induced flow of ϕ \phi on G / Γ G/\Gamma is assumed to be ergodic. Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say ϕ \phi is isolated if and only if for any ϕ ′ \phi ’ which recurrently approaches ϕ \phi for positive and negative time, ϕ \phi equals ϕ ′ \phi ’ up to sense-preserving reparameterization. Theorem B(a) states that if G G is one-connected and nilpotent, or one-connected and solvable with exp: L ( G ) → G L(G) \to G a diffeomrophism, then every ϕ \phi of G G is isolated. Let G G be connected and semisimple and ϕ ( t ) = exp ⁡ ( t X ) \phi (t) = \exp (tX) . Then Theorem B(b) states that ϕ \phi is isolated, if [ X , Y ] = 0 [X,\,Y] = 0 and ad ⁡ ( Y ) \operatorname {ad} (Y) being semisimple imply that ad ⁡ ( Y ) \operatorname {ad} (Y) has some eigenvalue not pure imaginary and not zero. If G G has finite center, ϕ \phi is isolated if there is no compact connected subgroup in the centralizer of ϕ \phi .

Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let S be the ring of all complex-valued polynomials on X, regarded as a G-module in the obvious way, and let J ? S be the sub-ring of all G-invariant polynomials on X.

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.

Weyl quantization for semidirect products

Let $G$ be the semidirect product $V\rtimes \,K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\pi$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$. We construct an invariant symbolic calculus for $\pi$, under some technical hypothesis. We give some examples including the Poincaré group.

Let G be a connected Lie group, T a discrete subgroup, <f>: R - G a one-parameter subgroup, and </>* : (RxG/r) - G/T, where <j>* : (t, xT) h-> (4>(t)x)r, the G-induced flow.G/T is a homogeneous space.Since the book Flows on homogeneous spaces by Auslander Green, and Hahn [3], the ergodic theory and topological dynamics of these flows have been much studied because they provide interesting examples which can be understood in this algebraic setting.(See for example, Auslander [2], Brezin and Moore [5].)The equivalences studied have usually been parameter preserving, measure theoretic or topological maps.(See Parry [23], Walters [31], Ratner [27, 28], and Witte [32].)In this paper we study topological equivalences which do not preserve the parameter.We use results concerning the divergence of one-parameter subgroups.These results may have their own geometric interest.

1. Symmetries of vector spaces: 1.1. What is a symmetry? 1.2. Distance is fundamental 1.3. Groups of symmetries 1.4. Bilinear forms and symmetries of spacetime 1.5. Putting the pieces together 1.6. A broader view: Lie groups 2. Complex numbers, quaternions and geometry: 2.1. Complex numbers 2.2. Quaternions 2.3. The geometry of rotations of R3 2.4. Putting the pieces together 2.5. A broader view: octonions 3. Linearization: 3.1. Tangent spaces 3.2. Group homomorphisms 3.3. Differentials 3.4. Putting the pieces together 3.5. A broader view: Hilbert's fifth problem 4. One-parameter subgroups and the exponential map: 4.1. One-parameter subgroups 4.2. The exponential map in dimension one 4.3. Calculating the matrix exponential 4.4. Properties of the matrix exponential 4.5. Using exp to determine L(G) 4.6. Differential equations 4.7. Putting the pieces together 4.8. A broader view: Lie and differential equations 4.9. Appendix on convergence 5. Lie algebras: 5.1. Lie algebras 5.2. Adjoint maps { big `A' and small `a' 5.3. Putting the pieces together 5.4. A broader view: Lie theory 6. Matrix groups over other fields: 6.1. What is a field? 6.2. The unitary group 6.3. Matrix groups over finite fields 6.4. Putting the pieces together 6.5. A broader view of finite groups of Lie type and simple groups Appendix I. Linear algebra facts Appendix II. Paper assignment used at Mount Holyoke College Appendix III. Opportunities for further study Solutions to selected problems Bibliography.

If V is a finite-dimensional real or complex vector space, let GL(V ) denote the group of invertible linear transformations of V. If we choose a basis for V, we can identify GL(V ) with $$\mathsf{GL}(n; \mathbb{R})$$ or $$\mathsf{GL}(n; \mathbb{C})$$ . Any such identification gives rise to a topology on GL(V ), which is easily seen to be independent of the choice of basis. With this discussion in mind, we think of GL(V ) as a matrix Lie group. Similarly, we let gl(V ) = End(V ) denote the space of all linear operators from V to itself, which forms a Lie algebra under the bracket $$[X,Y ] = \mathit{XY } -\mathit{YX}$$ .representationof a Lie algebrarepresentationof a matrix Lie group

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

Let G be a matrix Lie group. Then, a finite-dimensional complex representation of G is a Lie group homomorphism $$\prod \;:\;G \to GL\left( {n;{\Bbb C}} \right)\;$$ (n ≥ 1) or, more generally, a Lie group homomorphism $$\prod \;:\;G \to GL\left( V \right),\;$$ where V is a finite-dimensional complex vector space (with dim(V) ≥ 1). A finite-dimensional real representation of G is a Lie group homomorphism Π of G into GL(n; ℝ) or into GL(V), where V is a finite-dimensional real vector space.KeywordsIrreducible RepresentationInvariant SubspaceUnitary RepresentationHeisenberg GroupComplex Vector SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.