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Langer and Perline proved that if x is a solution of the geometric Airy curve flow on R^n then there exists a parallel normal frame along x(. ,t) for each t such that the corresponding principal curvatures satisfy the (n-1) component modified KdV (vmKdV_n). They also constructed higher order curve flows whose principal curvatures are solutions of the higher order flows in the vmKdV_n soliton hierarchy. In this paper, we write down a Poisson structure on the space of curves in R^n parametrized by the arc-length, show that the geometric Airy curve flow is Hamiltonian, write down a sequence of commuting Hamiltonians, and construct Backlund transformations and explicit soliton solutions., Comment: 27 pages

The $\hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $\hat B_n^{(1)}$, the Drinfeld-Sokolov $\hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $\hat B_n^{(1)}$-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) use loop group factorization to construct Darboux transforms (DTs) for the $\hat B_n^{(1)}$-hierarchy, (2) give a Permutability formula and scaling transform for these DTs, (3) use DTs of the $\hat B_{n}^{(1)}$-hierarchy to construct DTs for the $\hat B_n^{(1)}$-KdV and the isotropic curve flows of B-type, (4) give algorithm to construct soliton solutions and write down explicit soliton solutions for the third $\hat B_1^{(1}$-KdV, $\hat B_2^{(1)}$-KdV flows and isotropic curve flows on $\mathbb{R}^{2,1}$ and $\mathbb{R}^{3,2}$ of B-type., Comment: Comments are welcome

Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Chern's conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres., Comment: 25 pages, no figure. Added a reference and revised conjecture 4.18

Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= \sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $\gamma: R\to R^{n+1,n}$ is {\it isotropic} if $\gamma, \gamma_x, \ldots, \gamma_x^{(2n)}$ are linearly independent and the span of $\gamma, \ldots, \gamma_x^{(n-1)}$ is isotropic. Given an isotropic curve, we show that there is a unique up to translation parameter such that $(\gamma_x^{(n)}, \gamma_x^{(n)})=1$ (we call such parameter the isotropic parameter) and there also exists a natural moving frame. In this paper, we consider two sequences of curve flows on the space of isotropic curves parametrized by isotropic parameter. We show that differential invariants of these isotropic curves satisfy Drinfeld-Sokolov's KdV type soliton hierarchies associated to the affine Kac-Moody algebra $\hat B_n^{(1)}$ and $\hat A_{2n}^{(2)}$ Then we use techniques from soliton theory to construct bi-Hamiltonian structure, conservation laws, Backlund transformations and permutability formulas for these curve flows., Comment: 37 pages

We construct B\"acklund transformations (BT) for the Gelfand-Dickey hierarchy (GD$_n$-hierarchy) on the space of $n$-th order differential operators on the line. Suppose $L=\partial_x^n-\sum_{i=1}^{n-1}u_i\partial_x^{(i-1)}$ is a solution of the $j$-th GD$_n$ flow. We prove the following results: (1) There exists a system (BT)$_{u,k}$ of non-linear ordinary differential equations for $h:R^2\to C$ depending on $u_1, \ldots, u_{n-1}$ in $x$ and $t$ variables such that $\tilde L= (\partial+h)^{-1}L(\partial+h)$ is a solution of the $j$-th GD$_n$ flow if and only if $h$ is a solution of (BT)$_{u,k}$ for some parameter $k$. Moreover, coefficients of $\tilde L$ are differential polynomials of $u$ and $h$. We say such $\tilde L$ is obtained from a BT with parameter $k$ from $L$. (2) (BT)$_{u,k}$ is solvable. (3) There exists a compatible linear system for $\phi:R^2\to C$ depending on a parameter $k$, such that if $\phi_1, \ldots, \phi_{n-1}$ are linearly independent solutions of this linear system then $h:=(\ln W(\phi_1, \ldots, \phi_{n-1}))_x$ is a solution of (BT)$_{u,k}$ and $(\partial+h)^{-1} L (\partial+h)$ is a solution of the $j$-th GD$_n$ flow, where $W(\phi_1,\ldots,\phi_{n-1})$ is the Wronskian Moreover, these give all solutions of (BT)$_{u,k}$. (4) We show that the BT for the GD$_n$ hierarchy constructed by M. Adler is our BT with parameter $k=0$. (5) We construct a permutability formula for our BTs and infinitely many families of explicit rational solutions and soliton solutions., Comment: 29 pages

We construct a sequence of commuting central affine curve flows on $R^n\backslash 0$ invariant under the action of $SL(n,R)$ and prove the following results: (a) The central affine curvatures of a solution of the j-th central affine curve flow is a solution of the j-th flow of Gelfand-Dickey (GD$_n$) hierarchy on the space of n-th order differential operators. (b) We use the solution of the Cauchy problems of the GD$_n$ flow to solve the Cauchy problems for the central affine curve flows with periodic initial data and also with initial data whose central affine curvatures are rapidly decaying. (c) We obtain a bi-Hamiltonian structure for the central affine curve flow hierarchy and prove that it arises naturally from the Poisson structures of certain co-adjoint orbits. (d) We construct Backlund transformations, infinitely many families of explicit solutions and give a permutability formula for these curve flows., Comment: 32 pages (this version adds a section on Backlund transformations and makes some changes of the first version)

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flow is integrable if the evolution equation of the local differential invariants of a solution of the curve flow is a soliton equation. For example, the Hodge star mean curvature flow on $R^3$ and on $R^{2,1}$, the geometric Airy flow on $R^n$, the Schrodingier flow on compact Hermitian symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a compact Lie group are integrable. In this paper, we give a survey of these results, describe a systematic method to construct integrable curve flows from Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the Cauchy problem of these curve flows., Comment: 51 pages

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function tau_f for the hierarchy. In this third paper, we prove the following results for the nxn KdV hierarchy: (1) The second partials of ln(tau_f) are differential polynomials of the formal inverse scattering solution u_f. Moreover, u_f can be recovered from the second partials of ln(tau_f). (2) The natural Virasoro action on ln(tau_f) constructed in the second paper is given by partial differential operators in ln(tau_f). (3) There is a bijection between phase spaces of the nxn KdV hierarchy and the Gelfand-Dickey (GD_n) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the nxn KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD_n hierarchy under the bijection., Comment: 38 pages

We give the following results for Pinkall's central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data., Comment: to appear in Journal of Fixed Point Theory and Applications (Choquet-Bruhat Festschrift)

There is a general method for constructing a soliton hierarchy from a splitting of a loop group as a positive and a negative sub-groups together with a commuting linearly independent sequence in the positive Lie subalgebra. Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup a solution u_f of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function tau_f for each element f in the negative subgroup. In this paper, we give integral formulas for variations of ln(tau_f) and second partials of ln(tau_f), discuss whether we can recover solutions u_f from tau_f, and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the GL(n,\C)-hierarchy., Comment: 37 pages

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting positive elements, (ii) a method---based on (i)---for constructing soliton hierarchies from a symmetric space, (iii) the dressing action of the negative loop subgroup on the space of solutions of the related soliton equation, (iv) classical B\"acklund, Christoffel, Lie, and Ribaucour transformations for surfaces in three-space and their relation to dressing actions, (v) methods for constructing a Lax pair for the Gauss-Codazzi Equation of certain submanifolds that admit Lie transforms, (vi) how soliton theory can be used to generalize classical soliton surfaces to submanifolds of higher dimension and co-dimension.

We introduce two families of soliton hierarchies: the twisted hierarchies associated to symmetric spaces. The Lax pairs of these two hierarchies are Laurent polynomials in the spectral variable. Our constructions gives a hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE), which is the Gauss-Codazzi equation for n-dimensional submanifolds in Euclidean (2n-1)-space with constant sectional curvature -1. In fact, the GSGE is the first order system associated to a twisted Grassmannian system. We also study symmetries for the GSGE.

A diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is isothermic_k if it admits line of curvature co-ordinates such that its induced metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k orthogonal co-ordinate systems on R^n are invariant under conformal transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1} (Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems resp.) are related by Combescure transformations. We give a correspondence between Combescure sequences of Guichard_k orthogonal co-ordinate systems on R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable systems. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries., Comment: 21 pages, reworked definition to correct error in theorem

E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and constant multiplicities., Comment: 24 pages, 1 figure. Minor changes

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing., Comment: 24 pages, PDF file

The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat Lagrangian submanifolds in C^n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated to SU(n). In this article, we explain the method developed jointly with K. Uhlenbeck, that uses various loop group factorizations to construct inverse scattering transforms, Backlund transformations, and solutions to Cauchy problems for these equations.

We introduce a new integrable system hierarchy which is a restriction of the AKNS nxn hierarchy coming from an unusual splitting of the loop algebra. This splitting comes from an automorphism of the loop algebra instead of an automorphism of SL(n,C). It is known that the 2x2 KdV is the standard KdV hierarchy., Comment: We move section 4 and 5 of version 1 to a new paper titled "Tau function and Virasoro action for the nxn KdV hierarchy"

We associate a natural $\lambda$-family ($\lambda \in \R \setminus \{0\} $) of flat Lagrangian immersions in $\C^n$ with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space $\frac{\U(n) \ltimes \C^n}{\OO(n) \ltimes \R^n}$. An interesting observation is that the family degenerates to an Egoroff net on $\R^n$ when $\lambda \to 0$. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The action of the generator with one simple pole gives the geometric Ribaucour transformation and we provide the permutability formula for such transformations. The action of the generator with two poles and the action of a rational loop in the translation subgroup produce new transformations. The corresponding results for flat Lagrangian submanifolds in $\C P^{n-1}$ and $\p$-invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration., Comment: 23 pages, submitted

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang-Mills equation on $\R^{2,2}$. A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas-Ioannidou, Villarroel and Zakhorov-Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use B\"acklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data., Comment: 31 pages

We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit 6-sphere. 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 is the equation for primitive maps associated to the 6-symmetric space G_2/T^2, and use this to explain some of the known results. Moreover, the equation for S^1-symmetric almost complex curves in S^6 is the periodic Toda lattice associated to G_2, and a discussion of periodic solutions is given., Comment: to appear in Chinese Annals of Mathematics (2006), a special issue in memory of S. S. Chern

Let $\sigma$ be an involution of a real semi-simple Lie group $U$, $U_0$ the subgroup fixed by $\sigma$, and $U/U_0$ the corresponding symmetric space. Ferus and Pedit called a submanifold $M$ of a rank $r$ symmetric space $U/U_0$ a {\it curved flat} if $T_pM$ is tangent to an $r$-dimensional flat of $U/U_0$ at $p$ for each $p\in M$. They noted that the equation for curved flats is an integrable system. Bryant used the involution $\sigma$ to construct an involutive exterior differential system $\ci_\sigma$ such that integral submanifolds of $\ci_\sigma$ are curved flats. Terng used $r$ first flows in the $U/U_0$-hierarchy of commuting soliton equations to construct the $U/U_0$-system. She showed that the $U/U_0$-system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many these systems occur as the Gauss-Codazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system $\ci_\sigma$., Comment: 19 pages

The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a limiting method to construct 2-solitons whose extended solutions have a double pole. Ioannidou and Zakrzewski, and Anand constructed more soliton solutions whose extended solutions have a double or triple pole. Some of the main results of this paper are: (i) We construct algebraic B\"acklund transformations (BTs) that generate new solutions of the Ward equation from a given one by an algebraic method. (ii) We use an order $k$ limiting method and algebraic BTs to construct explicit Ward solitons, whose extended solutions have arbitrary poles and multiplicities. (iii) We prove that our construction gives all solitons of the Ward equation explicitly and the entries of Ward solitons must be rational functions in $x, y$ and $t$. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all $k$-unitons from finitely many rational maps from $C$ to $C^n$., Comment: 38 pages

We use algebraic Backlund transformations (BTs) to construct explicit solutions of the modified 2+1 chiral model from $T^2\times R$ to SU(n), where $T^2$ is a 2-torus. Algebraic BTs are parameterized by $z\in C$ (poles) and holomorphic maps $\pi$ from $T^2$ to Gr$(k,C^n)$. We apply B\"acklund transformations with carefully chosen poles and $\pi$'s to construct infinitely many solutions of the 2+1 chiral model that are (i) doubly periodic in space variables and periodic in time, i.e., triply periodic, (ii) homoclinic in the sense that the solution $u$ has the same stationary limit $u_0$ as $t\to \pm\infty$ and is tangent to a stable linear mode of $u_0$ as $t\to\infty$ and is tangent to an unstable mode of $u_0$ as $t\to -\infty$., Comment: 17 pages

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact Lie group $G$ and wave maps into $G$. We use B\"acklund transformations to construct explicit $2k$-soliton breather solutions for the -1 flow equation and show that the corresponding wave maps are periodic and homoclinic. The compact symmetric space $G/K$ can be embedded as a totally geodesic submanifold of $G$ via the Cartan embedding. We prescribe the constraint condition for the -1 flow equation associated to $G$ which insures that the corresponding wave map into $G$ actually lies in $G/K$. For example, when $G/K=SU(2)/SO(2)=S^2$, the constrained -1-flow equation associated to SU(2) has the sine-Gordon equation (SGE) as a subequation and classical breather solutions of the SGE are 2-soliton breathers. Thus our result generalizes the result of Shatah and Strauss that a classical breather solution of the SGE gives rise to a periodic homoclinic wave map to $S^2$. When the group $G$ is non-compact, the bi-invariant metric on $G$ is pseudo-Riemannian and B\"acklund transformations of a smooth solution often are singular. We use B\"acklund transformations to show that there exist smooth initial data with constant boundary conditions and finite energy such that the Cauchy problem for wave maps from $R^{1,1}$ to the pseudo-Riemannian manifold $SL(2,R)$ develops singularities in finite time., Comment: 32 pages

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schr\"odinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation for harmonic maps from the plane to a compact Lie group, for primitive maps from the plane to a $k$-symmetric space, and constant mean curvature surfaces and isothermic surfaces in space forms are integrable elliptic systems. We also give a survey of (i) construction of solutions using loop group factorizations, (ii) PDEs in differential geometry that are soliton equations or elliptic integrable systems, (iii) similarities and differences of soliton equations and integrable elliptic systems., Comment: 67 pages, to appear in Surveys on Geometry and Integrable Systems, Advanced Studies in Pure Mathematics, Mathematical Society of Japan

It is known that the Schr\"odinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schr\"odinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint $U$-orbits from flows in the soliton hierarchy associated to a compact Lie group $U$. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples., Comment: 25 pages

There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n variables, the so called U/K-system. Let G_{m,n} denote the Grassmannian of n-dimensional linear subspaces in R^{m+n}, and G_{m,n}^1 the Grassmannian of space like m-dimensional linear subspaces in the Lorentzian space R^{m+n,1}. In this paper, we use techniques from soliton theory to study submanifolds in space forms whose Gauss-Codazzi equations are gauge equivalent to the G_{m,n}-system or the G_{m,n}^1-system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G_{m,n} and G_{m,n}^1 systems correspond to B\"acklund, Darboux and Ribaucour transformations for submanifolds., Comment: 104 pages, 7 figures

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrodinger equation (MNLS). We construct via gauge transformations an isomorphism from C(R,M) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows follows from the correspondence. Direct geometric constructions show the flows are given by geometric partial differential equations, and the space of conservation laws has a structure of a non-abelian Poisson group. We also construct a hierarchy of symplectic structures for GNLS. Under pullback, the known order k symplectic structures correspond to the order (k-2) symplectic structures that we find. The shift by two is a surprise, and is due to the fact that the group structures depend on gauge choice., Comment: 29 pages

We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local B\"acklund transformations, and we derive a permutability formula from different factorizations of a quadratic element. We prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities. However, the action of the subgroup of the rational maps that satisfy the U(n)-reality condition $g(\bar\l)^*g(\l)=I$ on the space of global rapidly decaying solutions of the flows in the $u(n)$-hierarchy is global, and the action of a simple element gives a global B\"acklund transformation. The actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multi-solitons (multi-breathers). We show that this theory generalizes the classical B\"acklund theory of the sine-Gordon equation. The group structures of B\"acklund transformations for various hierarchies are determined by their reality conditions. We identify the reality conditions (the group structures) for the $sl(n,R)$, $u(k,n-k)$, KdV, Kupershmidt-Wilson, and Gel'fand-Dikii hierarchies. The actions of linear fractional transformations that satisfies a reality condition, modulo the center of the group of rational maps, gives B\"acklund and Darboux transformations for the hierarchy defined by the reality condition. Since the factorization cannot always be carried out under these reality condition, the action is again local, and B\"acklund transformations only generate local solutions for these hierarchies unless singular solutions are allowed., Comment: Plain Tex, 80 pages

Conservation laws, heirarchies, scattering theory and B\"acklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schr\"odinger equation, modified KdV, and the n-wave equation). We also discuss a number of applications in geometry, including the sine-Gordon equation, harmonic maps, Schr\"odinger flows on Hermitian symmetric spaces, Darboux orthogonal coordinates, and isometric immerisons of one space form in another., Comment: 85 pages