Geometric Optimization Methods for Adaptive Filtering

Acknowledgments

I would like to thank my advisor Professor Roger Brockett for directing me in three directions that eventually coalesced: subspace tracking, gradient flows on Lie groups, and conjugate gradient methods on symmetric spaces. I am also indebted to Tony Bloch for the invitation to speak at the Fields Institute workshop on Hamiltonian and gradient flows in April 1992, and to the Fields Institute itself for their generous support during my visit. I greatly appreciated the opportunity to present my work to the distinguished participants of this workshop. My exposition was influenced by the teaching style of Professor Guillemin, and I benefited from many helpful discussions with Professor Anderson, who has always been generous with his time and advice. I would also like to thank the members of my thesis committee: Professors James Clark, Petros Maragos, and David Mumford, not only for their evaluation of this work, but also for the intellectual and social qualities that they and their students brought to the Harvard Robotics Laboratory.

My life in graduate school would have been much more difficult without the pleasant and supportive atmosphere that my classmates provided. The help and advice of my predecessors was invaluable in learning the graduate student ropes. I will always remember sitting on the bank of Walden Pond with Bill Nowlin, Nicola Ferrier, and Ann Stokes on my first day off after a very long first year. “I’m supposed to pick a project this summer,” I said. Bill was quick with his advice: “The summer after first year is just a waste.” I also remember Nicola bringing cookies to me and Bob Hewes late one night because she remembered how heavy the work load was. Ken Keeler and Frank Park enriched the cultural life of Pierce G14 by founding an ever growing museum. Peter Hallinan and Tai Sing Lee taught me some climbing techniques at Acadia Natl. Park. Gaile Gordon hosted many enjoyable and memorable social events. Ed Rak convinced me that I could hack TeX macros and still graduate. Ann Stokes’s friendship made office life enjoyable even in the batcave. John Page and George Thomas worked very hard to ensure that the lab ran smoothly. Leonid Faybusovich taught me about the gradient. I thoroughly enjoyed the discussions I shared with Dan Friedman, Jeff Kosowsky, Bob Hewes, Peter Belhumeur, and others. We all owe a debt to Navin Saxena for making our Wednesday afternoon lab tea a thriving event.

Finally, I thank my family, especially my parents for their encouragement, empathy, and love. Most of all, I thank my wife Laura, whom I love very much. Her constant thoughtfulness, support, patience, and love made many difficult days good, and many good days great.

Abstract

The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization problem. Some classical optimization techniques on Euclidean space are generalized to Riemannian manifolds. Several algorithms are presented and their convergence properties are analyzed employing the Riemannian structure of the manifold. Specifically, two new algorithms, which can be thought of as Newton’s method and the conjugate gradient method on Riemannian manifolds, are presented and shown to possess quadratic and superlinear convergence, respectively. These methods are applied to several eigenvalue and singular value problems, which are posed as constrained optimization problems. New efficient algorithms for the eigenvalue problem are obtained by exploiting the special homogeneous space structure of the constraint manifold. It is shown that Newton’s method applied to the Rayleigh quotient on a sphere converges cubically, and that the Rayleigh quotient iteration is an efficient approximation of Newton’s method. The Riemannian version of the conjugate gradient method applied to this function gives a new algorithm for finding the eigenvectors corresponding to the extreme eigenvalues of a symmetric matrix. The Riemannian version of the conjugate gradient method applied to a generalized Rayleigh quotient yields a superlinearly convergent algorithm for computing the $k$ eigenvectors corresponding to the extreme eigenvalues of an $n$-by-$n$ matrix. This algorithm requires $O(nk^{2})$ operations and $O(k)$ matrix-vector multiplications per step. Several gradient flows are analyzed that solve eigenvalue and singular value problems. The new optimization algorithms are applied to the subspace tracking problem of adaptive signal processing. A new algorithm for subspace tracking is given, which is based upon the conjugate gradient method applied to the generalized Rayleigh quotient. The results of several numerical experiments demonstrating the convergence properties of the new algorithms are given.

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Chapter 1 Introduction

Optimization is the central idea behind many problems in science and engineering. Indeed, determination of “the best” is both a practical and an aesthetic problem that is encountered almost universally. Thus it is not surprising to find in many areas of study a variety of optimization methods and vocabulary. While the statement of the optimization problem is simple—given a set of points and an assignment of a real number to each point, find the point with the largest or smallest number—its solution is not. In general, the choice of optimization algorithm depends upon many factors and assumptions about the underlying set and the real-valued function defined on the set. If the set is discrete, then a simple search and comparison algorithm is appropriate. If the discrete set is endowed with a topology, then a tree searching algorithm can yield a local extremum. If the set is a finite-dimensional vector space and the function is continuous, then the simplex method can yield a local extremum. If the set is a Euclidean space, i.e., a finite-dimensional vector space with inner product, and the function is differentiable, then gradient-based methods may be used. If the set is a polytope, i.e., a subset of Euclidean space defined by linear inequality constraints, and a linear function, then linear programming techniques are appropriate. This list indicates how successful optimization techniques exploit the given structure of the underlying space. This idea is an important theme of this thesis, which explains how the metric structure on a manifold may be used to develop effective optimization methods on such a space.

Manifolds endowed with a metric structure, i.e., Riemannian manifolds, arise naturally in many applications involving optimization problems. For example, the largest eigenvalue of a symmetric matrix corresponds to the point on a sphere maximizing the Rayleigh quotient. This eigenvalue problem and its generalizations are encountered in diverse fields: signal processing, mechanics, control theory, estimation theory, and others. In most cases the so-called principal invariant subspace of a matrix must be computed. This is the subspace spanned by the eigenvectors or singular vectors corresponding to the largest eigenvalues or singular values, respectively. Oftentimes there is an adaptive context so that the principal invariant subspaces change over time and must be followed with an efficient tracking algorithm. Many algorithms rely upon optimization techniques such as gradient following to perform this tracking.

A few analytic optimization methods are quite old, but, as in most computational fields, the invention of electronic computers was the impetus for the development of modern optimization theory and techniques. Newton’s method has been a well-known approach for solving optimization problems of one or many variables for centuries. The method of steepest descent to minimize a function of several variables goes back to Cauchy. Its properties and performance are well-known; see, e.g., the books of ?, ?, or ? for a description and analysis of this technique. Modern optimization algorithms appeared in the middle of this century, with the introduction of linear and quadratic programming algorithms, the conjugate gradient algorithm of ?, and the variable metric algorithm of ?. It is now understood how these algorithms may be used to compute the point in ${\bf R}^{n}$ at which a differentiable function attains its maximum value, and what performance may be expected of them.

Of course, not all optimization problems are posed on a Euclidean space, and much research has been done on the constrained optimization problem, specifically when the underlying space is defined by equality constraints on Euclidean space. Because all Riemannian manifolds may be defined in this way, this approach is general enough for the purposes of this thesis. What optimization algorithms are appropriate on such a space? ? considers this question in his exposition of the constrained optimization problem. He describes an idealized steepest descent algorithm on the constraint surface that employs geodesics in gradient directions, noting that this approach is in general not computationally feasible. For this reason, other approaches to the constrained optimization problem have been developed. All of these methods depend upon the imbedding of the constraint surface in ${\bf R}^{n}$. Projective methods compute a gradient vector tangent to the constraint surface, compute a minimum in ${\bf R}^{n}$ along this direction, then project this point onto the constraint surface. Lagrange multiplier methods minimize a function defined on ${\bf R}^{n}$ constructed from the original function to be minimized and the distance to the constraint surface. However, this so-called extrinsic approach ignores the intrinsic structure that the manifold may have. With specific examples, such as a sphere and others to be discussed later, intrinsic approaches are computationally feasible, but the study of intrinsic optimization algorithms is absent from the literature.

Optimization techniques have long been applied to the fields of adaptive filtering and control. There is a need for such algorithms in these two fields because of their reliance on error minimization techniques and on the minimax characterization of the eigenvalue problem. Also, many scenarios in adaptive filtering and control have slowly varying parameters which corresponding to the minimum point of some function that must be estimated and tracked. Gradient-based algorithms are desirable in this situation because the minimum point is ordinarily close to the current estimate, and the gradient provides local information about the direction of greatest decrease.

Many researchers have applied constrained optimization techniques to algorithms that compute the static or time varying principal invariant subspaces of a symmetric matrix. This problem may be viewed as the problem of computing $k$ orthonormal vectors in ${\bf R}^{n}$ that maximize a generalized form of the Rayleigh quotient. Orthonormality imposes the constraint surface. ? propose a projective formulation of the constrained conjugate gradient method to solve the symmetric eigenvalue problem. ? proposes a very similar method for application to finite element eigenvalue problems. ? are the first to apply this projective conjugate gradient method to the problem of adaptive spectral estimation for signal processing. However, these conjugate gradient algorithms are based upon the classical unconstrained conjugate gradient method on Euclidean space. They apply this algorithm to the constrained problem without accounting for the curvature terms that naturally arise. In general, the superlinear convergence guaranteed by the classical conjugate gradient method is lost in the constrained case when these curvature terms are ignored.

? recognize this fact in their constrained conjugate gradient algorithm for maximizing the Rayleigh quotient on a sphere. They correctly utilize the curvature of the sphere to develop a conjugate gradient algorithm on this space analogous to the classical superlinearly convergent conjugate gradient algorithm. Insofar as they use maximization along geodesics on the sphere instead of maximization along lines in ${\bf R}^{n}$ followed by projection, their approach is the first conjugate gradient method employing instrinsic ideas to appear. However, they use an azimuthal projection to identify points on the sphere with points in tangent planes, which is not naturally defined because it depends upon the choice of imbedding. Thus their method is extrinsic. Although the asymptotic performance of their constrained conjugate gradient algorithm is the same as one to be presented in this thesis, their dependence on azimuthal projection does not generalize to other manifolds. We shall see that completely intrinsic approaches on arbitrary Riemannian manifolds are possible and desirable.

There are many other algorithms for computing the principal invariant subspaces that are required for some methods used in adaptive filtering [ComonGolub]. Of course, one could apply the QR algorithm at each step in the adaptive filtering procedure to obtain a full diagonal decomposition of a symmetric matrix, but this requires $O(n^{3})$ floating point operations ($n$ is the dimension of the matrix), which is unnecessarily expensive. Also, many applications require only the principal invariant subspace corresponding to the $k$ largest eigenvalues, thus a full decomposition involves wasted effort. Furthermore, this technique does not exploit previous information, which is important in most adaptive contexts. So other techniques for obtaining the eigenvalue decomposition are used. In addition to the constrained conjugate gradient approaches mentioned in the preceding paragraphs, pure gradient based methods and other iterative techniques are popular. The use of gradient techniques in adaptive signal processing was pioneered in the 1960s. See ? for background and references. Several algorithms for the adaptive eigenvalue problem use such gradient ideas [Owsley, Larimore, Hu].

Iterative algorithms such as Lanczos methods are very important in adaptive subspace tracking problems. Lanczos methods compute a sequence of tridiagonal matrices (or bidiagonal matrices in the case where singular vectors are required) whose eigenvalues approximate the extreme eigenvalues of the original matrix. The computational requirements of the classical Lanczos algorithm are modest: only $O(nk^{2})$ operations and $O(k)$ matrix-vector multiplications are required to compute $k$ eigenvectors of an $n$-by-$n$ symmetric matrix. Thus Lanczos methods are well-suited for sparse matrix extreme eigenvalue problems. However, the convergence properties of the classical Lanczos methods are troublesome, and they must be modified to yield useful algorithms [ParlettScott, Parlett, GVL, CullumWill].

This thesis arose from the study of gradient flows applied to the subspace tracking problem as described by ?, and from the study of gradient flows that diagonalize matrices [Brockett:sort]. While the resulting differential equation models are appealing from the perspective of learning theory, it is computationally impractical to implement them on conventional computers. A desire to avoid the “small step” methods found in the integration of gradient flows while retaining their useful optimization properties led to the investigation of “large step” methods on manifolds, analogous to the optimization algorithms on Euclidean space discussed above. A theory of such methods was established, and then applied to the subspace tracking problem, whose homogeneous space structure allows efficient and practical optimization algorithms.

The following contributions are contained within this thesis. In Chapter 2, a geometric framework is provided for a large class of problems in numerical linear algebra. This chapter reviews the natural metric structure of various Lie groups and homogeneous spaces, along with some useful formulae implied by this structure, which will be used throughout the thesis. This geometric framework allows one to solve problems in numerical linear algebra, such as the computation of eigenvalues and eigenvectors, and singular values and singular vectors, with gradient flows on Lie groups and homogeneous spaces.

In Chapter 3 a gradient flow that yields the extreme eigenvalues and corresponding eigenvectors of a symmetric matrix is given, together with a gradient flow that yields the singular value decomposition of an arbitrary matrix. (Functions whose gradient flows yield the extreme singular values and corresponding left singular vectors of an arbitrary matrix are also discussed in Chapter 5.)

Chapter 4 develops aspects of the theory of optimization of differentiable functions defined on Riemannian manifolds. New methods and a new point of view for solving constrained optimization problems are provided. Within this chapter, the usual versions of Newton’s method and the conjugate gradient method are generalized to yield new optimization algorithms on Riemannian manifolds. The method of steepest descent on a Riemannian manifold is first analyzed. Newton’s method on Riemannian manifolds is developed next and a proof of quadratic convergence is given. The conjugate gradient method is in then introduced with a proof of superlinear convergence. Several illustrative examples are offered throughout this chapter. These three algorithms are applied to the Rayleigh quotient defined on the sphere, and the function $f(\Theta)=\mathop{\rm tr}\nolimits\Theta^{\scriptscriptstyle\rm T}Q\Theta N$ defined on the special orthogonal group. It is shown that Newton’s method applied to the Rayleigh quotient converges cubically, and that this procedure is efficiently approximated by the Rayleigh quotient iteration. The conjugate gradient algorithm applied to the Rayleigh quotient on the sphere yields a new superlinearly convergent algorithm for computing the eigenvector corresponding to the extreme eigenvalue of a symmetric matrix, which requires two matrix-vector multiplications and $O(n)$ operations per iteration.

Chapter 5 applies the techniques developed in the preceding chapters to the subspace tracking problem of adaptive signal processing. The idea of tracking a principal invariant subspace is reviewed in this context, and it is shown how this problem may be viewed as maximization of the generalized Rayleigh quotient on the so-called Stiefel manifold of matrices with orthonormal columns. An efficient conjugate gradient method that solves this optimization problem is developed next. This algorithm, like Lanczos methods, requires $O(nk^{2})$ operations per step and $O(k)$ matrix-vector multiplications. This favorable computational cost is dependent on the homogeneous space structure of the Stiefel manifold; a description of the algorithms implementation is provided. Superlinear convergence of this algorithm to the eigenvectors corresponding to the extreme eigenvalues of a symmetric matrix is assured by results of Chapter 4. This algorithm also has the desirable feature of maintaining the orthonormality of the $k$ vectors at each step. A similar algorithm for computing the largest left singular vectors corresponding to the extreme singular values of an arbitrary matrix is discussed. Finally, this algorithm is applied to the subspace tracking problem. A new algorithm for subspace tracking is given, which is based upon the conjugate gradient method applied to the generalized Rayleigh quotient. The results of several numerical experiments demonstrating the tracking properties of this algorithm are given.

Chapter 2 Riemannian geometry of Lie groups and homogeneous spaces

Both the analysis and development of optimization algorithms presented in this thesis rely heavily upon the geometry of the space on which optimization problems are posed. This chapter provides a review of pertinent ideas from differential and Riemannian geometry, Lie groups, and homogeneous spaces that will be used throughout the thesis. It may be skipped by those readers familiar with Riemannian geometry. Sections 1 and 2 contain the necessary theoretical background. Section 3 provides formulae specific to the manifolds to be used throughout this thesis, which are derived from the theory contained in the previous sections.

In this section the concepts of Riemannian structures, affine connections, geodesics, parallel translation, and Riemannian connections on a differentiable manifold are reviewed. A background of differentiable manifolds and tensor fields is assumed, e.g., Chapters 1–5 of ? or the introduction of ?. The review follows Helgason’s (?) and Spivak’s (?, Vol. 2) expositions.

Let $M$ be a $C^{\infty}$ differentiable manifold. Denote the set of $C^{\infty}$ functions on $M$ by $C^{\infty}(M)$, the tangent plane at $p$ in $M$ by $T_{p}$ or $T_{p}M$, and the set of $C^{\infty}$ vector fields on $M$ by ${X}(M)$.

Let $M$ be a differentiable manifold. A Riemannian structure on $M$ is a tensor field $g$ of type $(0,2)$ which for all $X$, $Y\in{X}(M)$ and $p\in M$ satisfies

A Riemannian manifold is a connected differentiable manifold with a Riemannian structure. For every $p$ in $M$, the Riemannian structure $g$ provides an inner product on $T_{p}$ given by the nondegenerate symmetric bilinear form $g_{p}\colon T_{p}\times T_{p}\to{\bf R}$. The notation $\langle X,Y\rangle=g_{p}(X,Y)$ and $\|X\|=g_{p}(X,X)^{1/2}$, where $X$, $Y\in T_{p}$, is often used. Let $t\mapsto\mathchar 28941\relax(t)$, $t\in[a,b]$, be a curve segment in $M$. The length of $\mathchar 28941\relax$ is defined by the formula

Because $M$ is connected, any two points $p$ and $q$ in $M$ can be joined by a curve. The infimum of the length of all curve segments joining $p$ and $q$ yields a metric on $M$ called the Riemannian metric and denoted by $d(p,q)$.

Let $M$ be a Riemannian manifold with Riemannian structure $g$ and $f\colon M\to{\bf R}$ a $C^{\infty}$ function on $M$. The gradient of $f$ at $p$, denoted by $(\mathop{\rm grad}\nolimits{\!f})_{p}$, is the unique vector in $T_{p}$ such that $df_{p}(X)=\langle(\mathop{\rm grad}\nolimits{\!f})_{p},X\rangle$ for all $X$ in $T_{p}$.

The corresponding vector field $\mathop{\rm grad}\nolimits{\!f}$ on $M$ is clearly smooth.

The expression of the preceding ideas using coordinates is often useful. Let $M$ be an $n\hbox{-}$dimensional Riemannian manifold with Riemannian structure $g$, and $(U,x^{1},\ldots,x^{n})$ a coordinate chart on $M$. There exist $n^{2}$ functions $g_{ij}$, $1\leq i,j\leq n$, on $U$ such that

Clearly $g_{ij}=g_{ji}$ for all $i$ and $j$. Because $g_{p}$ is nondegenerate for all $p\in U\subset M$, the symmetric matrix $(g_{ij})$ is invertible. The elements of its inverse are denoted by $g^{kl}$, i.e., $\sum_{l}g^{il}g_{lj}=\mathchar 28942\relax^{i}{}_{j}$, where $\mathchar 28942\relax^{i}{}_{j}$ is the Kronecker delta. Furthermore, given $f\in C^{\infty}(M)$, we have

Therefore, from the definition of $\mathop{\rm grad}\nolimits{\!f}$ above, we see that

Affine connections

Let $M$ be a differentiable manifold. An affine connection on $M$ is a function $\nabla$ which assigns to each vector field $X\in{X}(M)$ an ${\bf R}$-linear map $\nabla_{\!X}\colon{X}(M)\to{X}(M)$ which satisfies

for all $f\!$, $g\in C^{\infty}(M)$, $X$, $Y\in{X}(M)$. The map $\nabla_{\!X}$ may be applied to tensors of arbitrary type. Let $\nabla$ be an affine connection on $M$ and $X\in{X}(M)$. Then there exists a unique ${\bf R}$-linear map $A\mapsto\nabla_{\!X}A$ of $C^{\infty}$ tensor fields into $C^{\infty}$ tensor fields which satisfies

where $f\in C^{\infty}(M)$, $Y\in{X}(M)$, and $A$, $B$ are $C^{\infty}$ tensor fields. If $A$ is of type $(k,l)$, then $\nabla_{\!X}A$, called the covariant derivative of $A$ along $X$, is of type $(k,l)$, and ${\nabla\!A}\colon X\mapsto\nabla_{\!X}A$, called the covariant differential of $A$, is of type $(k,l+1)$.

The expression of these ideas using coordinates is useful. Let $M$ be an $n\hbox{-}$dimensional differentiable manifold with affine connection $\nabla$, and $(U,x^{1},\ldots,x^{n})$ a coordinate chart on $M$. These coordinates induce the canonical basis $(\mathchar 28992\relax/\mathchar 28992\relax x^{1})$, …, $(\mathchar 28992\relax/\mathchar 28992\relax x^{n})$ of ${X}(U)$. There exist $n^{3}$ functions $\Gamma_{ij}^{k}$, $1\leq i,j,k\leq n$, on $U$ such that

The $\Gamma_{ij}^{k}$ are called the Christoffel symbols of the connection.

The convergence proofs of later chapters require an analysis of the second order terms of real-valued functions near critical points. Consider the second covariant differential $\nabla\nabla{\!f}=\nabla^{2}{\!f}$ of a smooth function $f\colon M\to{\bf R}$. If $(U,x^{1},\ldots,x^{n})$ is a coordinate chart on $M$, then this $(0,2)$ tensor takes the form

Geodesics and parallelism

Let $M$ be a differentiable manifold with affine connection $\nabla$. Let $\mathchar 28941\relax\colon I\to M$ be a smooth curve with tangent vectors $X(t)=\dot{\mathchar 28941\relax}(t)$, where $I\subset{\bf R}$ is an open interval. The curve $\mathchar 28941\relax$ is called a geodesic if $\nabla_{\!X}X=0$ for all $t\in I$. Let $Y(t)\in T_{\mathchar 28941\relax(t)}$ ($t\in I$) be a smooth family of tangent vectors defined along $\mathchar 28941\relax$. The family $Y(t)$ is said to be parallel along $\mathchar 28941\relax$ if $\nabla_{\!X}Y=0$ for all $t\in I$.

For every $p$ in $M$ and $X\neq 0$ in $T_{p}$, there exists a unique geodesic t\mapsto\mathchar 28941\relax_{\lower 1.0pt\hbox{\scriptstyle X}}(t) such that \mathchar 28941\relax_{\lower 1.0pt\hbox{\scriptstyle X}}(0)=p and \dot{\mathchar 28941\relax}_{\lower 1.0pt\hbox{\scriptstyle X}}(0)=X. We define the exponential map $\exp_{p}\colon T_{p}\to M$ by \exp_{p}(X)=\mathchar 28941\relax_{\lower 1.0pt\hbox{\scriptstyle X}}(1) for all $X\in T_{p}$ such that $1$ is in the domain of \mathchar 28941\relax_{\lower 1.0pt\hbox{\scriptstyle X}}. Oftentimes the map $\exp_{p}$ will be denoted by “$\exp$” when the choice of tangent plane is clear, and \mathchar 28941\relax_{\lower 1.0pt\hbox{\scriptstyle X}}(t) will be denoted by $\exp tX$. A neighborhood $N_{p}$ of $p$ in $M$ is a normal neighborhood if $N_{p}=\exp N_{0}$, where $N_{0}$ is a star-shaped neighborhood of the origin in $T_{p}$ and $\exp$ maps $N_{0}$ diffeomorphically onto $N_{p}$. Normal neighborhoods always exist.

Given a curve $\mathchar 28941\relax\colon I\to M$ such that $\mathchar 28941\relax(0)=p$, for each $Y\in T_{p}$ there exists a unique family $Y(t)\in T_{\mathchar 28941\relax(t)}$ ($t\in I$) of tangent vectors parallel along $\mathchar 28941\relax$ such that $Y(0)=Y$. If $\mathchar 28941\relax$ joins the points $p$ and $\mathchar 28941\relax(\mathchar 28939\relax)=q$, the parallelism along $\mathchar 28941\relax$ induces an isomorphism $\mathchar 28956\relax_{pq}\colon T_{p}\to T_{q}$ defined by $\mathchar 28956\relax_{pq}Y=Y(\mathchar 28939\relax)$. If $\mathchar 28950\relax\in T_{p}^{*}$, define $\mathchar 28956\relax_{pq}\mathchar 28950\relax\in T_{q}^{*}$ by the formula $(\mathchar 28956\relax_{pq}\mathchar 28950\relax)(X)=\mathchar 28950\relax(\mathchar 28956\relax_{pq}^{-1}X)$ for all $X\in T_{q}$. The isomorphism $\mathchar 28956\relax_{pq}$ can be extended in an obvious way to mixed tensor products of arbitrary type.

Let $(U,x^{1},\ldots,x^{n})$ be a coordinate chart on an $n\hbox{-}$dimensional differentiable manifold with affine connection $\nabla$. Geodesics in $U$ satisfy the $n$ second order nonlinear differential equations

For example, geodesics on the imbedded $2$-sphere in ${\bf R}^{3}$ with respect to the connection given by $\Gamma_{ij}^{k}=\mathchar 28942\relax_{ij}x^{k}$ (the Levi-Civita connection on the sphere), $1\leq i,j,k\leq 3$, are segments of great circles, as shown in Figure 1. Let $t\mapsto\mathchar 28941\relax(t)$ be a curve in $U$, and let $Y=\sum_{k}Y^{k}\,(\mathchar 28992\relax/\mathchar 28992\relax x^{k})$ be a vector field parallel along $\mathchar 28941\relax$. Then the functions $Y^{k}$ satisfy the $n$ first order linear differential equations

For example, if $\mathchar 28941\relax$ is a segment of a great circle on the sphere, then parallel translation of vectors along $\mathchar 28941\relax$ with respect to the connection given by $\Gamma_{ij}^{k}=\mathchar 28942\relax_{ij}x^{k}$ is equivalent to rotating tangent planes along the great circle. The parallel translation of a vector tangent to the north pole around a geodesic triangle on $S^{2}$ is illustrated in Figure 1. Note that the tangent vector obtained by this process is different from the original tangent vector.

Parallel translation and covariant differentiation are related in the following way. Let $X$ be a vector field on $M$, and $t\mapsto\mathchar 28941\relax(t)$ an integral curve of $X$. Denote the parallelism along $\mathchar 28941\relax$ from $p=\mathchar 28941\relax(0)$ to $\mathchar 28941\relax(h)$, $h$ small, by $\mathchar 28956\relax_{h}$. Then for an arbitrary tensor field $A$ on $M$,

The covariant differentiation of a vector field $Y$ along a vector field $X$ is illustrated in Figure 1 at the left.

Riemannian connections

Given a Riemannian structure $g$ on a differentiable manifold $M$, there exists a unique affine connection $\nabla$ on $M$, called the Riemannian or Levi-Civita connection, which for all $X$, $Y\in{X}(M)$ satisfies

Length minimizing curves on $M$ are geodesics of the Levi-Civita connection. We shall use this connection throughout the thesis. For every $p\in M$, there exists a normal neighborhood $N_{p}=\exp N_{0}$ of $p$ such that $d(p,\exp_{p}X)=\|X\|$ for all $X\in N_{0}$, where $d$ is the Riemannian metric corresponding to $g$.

If $(U,x^{1},\ldots,x^{n})$ is a coordinate patch on $M$, then the Christoffel symbols $\Gamma_{ij}^{k}$ of the Levi-Civita connection are related to the functions $g_{ij}$ by the formula

By inspection it is seen that $\Gamma_{ij}^{k}=\Gamma_{ji}^{k}$.

Lie groups and homogeneous spaces

The basic structure of Lie groups and homogeneous spaces is reviewed in this section, which follows Helgason’s (?), Warner’s (?), Cheeger and Ebin’s (?), and Kobayashi and Nomizu’s (?, Chap. 10) expositions.

A Lie group $G$ is a differentiable manifold and a group such that the map $G\times G\to G$ defined by $(g,k)\mapsto gk^{-1}$ is $C^{\infty}$.

The identity in $G$ will be denoted by $e$ in the general case, and by $I$ if $G$ is a matrix group.

A Lie algebra ${g}$ over ${\bf R}$ is a vector space over ${\bf R}$ with a bilinear operation $[\,{,}\,]\colon{g}\times{g}\to{g}$ (called the bracket) such that for all $x$, $y$, $z\in{g}$,

Let $G$ be a Lie group and $g\in G$. Left multiplication by $g$ is denoted by the map $l_{g}\colon G\to G$, $k\mapsto gk$, and similarly for right multiplication $r_{g}\colon k\mapsto kg$. Let $X$ be a vector field on $G$. $X$ is said to be left invariant if for each $g\in G$,

For every element $X$ in ${g}$, there is a unique homomorphism $\mathchar 28958\relax\colon{\bf R}\to G$, called the one-parameter subgroup of $G$ generated by $X$, such that $\dot{\mathchar 28958\relax}(0)=X$. Define the exponential map $\exp\colon{g}\to G$ by setting $\exp X=\mathchar 28958\relax(1)$. The one-parameter subgroup $t\mapsto\mathchar 28958\relax(t)$ generated by $X$ is denoted by $t\mapsto\exp tX$. For matrix groups, the exponential map corresponds to matrix exponentiation, i.e., $\exp tX=e^{Xt}=I+tX+(t^{2}/2!)X^{2}+\cdots\,$. It will be seen in the next section in what sense the exponential map for a Lie group is related to the exponential map for a manifold with an affine connection.

Let $G$ be a Lie group with Lie algebra ${g}$. Consider the action of $G$ on itself by conjugation, i.e., $a\colon(g,k)\mapsto gkg^{-1}$, which has a fixed point at the identity. Denote the automorphism $k\mapsto gkg^{-1}$ of $G$ by $a_{g}$. Define the adjoint representation $\mathop{\rm Ad}\nolimits\colon G\to\mathord{\rm Aut}({g})$ by the map $g\mapsto(da_{g})_{e}$, where $\mathord{\rm Aut}({g})$ is the group of automorphisms of the Lie algebra ${g}$. If $G$ is a matrix group with $g\in G$ and $\mathchar 28961\relax\in{g}$, we have $\mathop{\rm Ad}\nolimits(g)(\mathchar 28961\relax)=g\mathchar 28961\relax g^{-1}$. Furthermore, we denote the differential of $\mathop{\rm Ad}\nolimits$ at the identity by $\mathop{\rm ad}\nolimits$, i.e.,

so that $\mathop{\rm ad}\nolimits\colon{g}\to\mathord{\rm End}({g})$ is a map from the Lie algebra ${g}$ to its vector space of endomorphisms $\mathord{\rm End}({g})$. The notation $\mathop{\rm Ad}\nolimits_{g}=\mathop{\rm Ad}\nolimits(g)$ ($g\in G$) and $\mathop{\rm ad}\nolimits_{X}=\mathop{\rm ad}\nolimits(X)$ ($X\in{g}$) is often used. It may be verified that $\mathop{\rm ad}\nolimits_{X}Y=[X,Y]$ for $X$ and $Y$ in ${g}$. If $G$ is a matrix group, then $\mathop{\rm ad}\nolimits_{X}Y=XY-YX$. The functions $\mathop{\rm Ad}\nolimits\colon G\to\mathord{\rm Aut}({g})$ and $\mathop{\rm ad}\nolimits\colon{g}\to\mathord{\rm End}({g})$ are related by

i.e., for $X\in{g}$, $\mathop{\rm Ad}\nolimits_{\exp X}=e^{\mathop{\rm ad}\nolimits_{X}}$.

Let ${g}$ be a Lie algebra. The Killing form of ${g}$ is the bilinear form $\mathchar 28967\relax$ on ${g}\times{g}$ defined by

Homogeneous spaces

Let $G$ be a Lie group and $H$ a closed subgroup of $G$. Then the (left) coset space $G/H=\{\,gH:g\in G\,\}$ admits the structure of a differentiable manifold such that the natural projection $\mathchar 28953\relax\colon G\to G/H$, $g\mapsto gH$, and the action of $G$ on $G/H$ defined by $(g,kH)\mapsto gkH$ are $C^{\infty}$. The dimension of $G/H$ is given by $\dim G/H=\dim G-\dim H$. Define the origin of $G/H$ by $o=\mathchar 28953\relax(e)$.

Let $G$ be a Lie group and $H$ a closed subgroup of $G$. The differentiable manifold $G/H$ is called a homogeneous space.

Let ${g}$ and ${h}$ be the Lie algebras of $G$ and $H$, respectively, and let ${m}$ be a vector subspace of ${g}$ such that ${g}={m}+{h}$ (direct sum). Then there exists a neighborhood of $0\in{m}$ which is mapped homeomorphically onto a neighborhood of the origin $o\in G/H$ by the mapping $\mathchar 28953\relax\circ\exp|_{m}$. The tangent plane $T_{o}(G/H)$ at the origin can be identified with the vector subspace ${m}$.

A Lie transformation group $G$ acting on a differentiable manifold $M$ is a Lie group $G$ which acts on $M$ (on the left) such that (i) every element $g\in G$ induces a diffeomorphism of $M$ onto itself, denoted by $p\mapsto g\cdot p$ or $p\mapsto l_{g}(p)$, (ii) the map from $G\times M$ to $M$ defined by $(g,p)\mapsto g\cdot p$ is $C^{\infty}$, and (iii) $g\cdot(k\cdot p)=gk\cdot p$ for $p\in M$, $g$, $k\in G$ (the action is transitive).

For example, the Lie group $G$ is clearly a Lie transformation group of the homogeneous space $G/H$.

The action of $G$ on $M$ is said to be effective if for any $g\in G$, $l_{g}=\mathop{\rm id}\nolimits$ on $M$ implies that $g=e$. Define the isotropy group $H_{p}$ at $p$ in $M$ by

The isotropy group at $p$ is a closed subgroup of $G$, and the mapping

Let $G/H$ be a homogeneous space of $G$, and $\mathchar 28953\relax\colon G\to G/H$ the natural projection. The tangent plane $T_{o}(G/H)$ at the origin $o=\mathchar 28953\relax(e)$ may be identified with the quotient space ${g}/{h}$, because for any function $f\in C^{\infty}(G/H)$,

where $\bar{f}$ is the unique lift in $C^{\infty}(G)$ such that $\bar{f}=f\circ\mathchar 28953\relax$. A tensor field $A$ on $G/H$ is $G$-invariant if and only if $A_{o}$ is invariant under the linear isotropy group at $o$, thus a computation of the map $l_{h}{}_{*}\colon T_{o}\to T_{o}$ is desirable. Let $\bar{l}_{g}\colon G\to G$ and $l_{g}\colon G/H\to G/H$ denote left translation by $g\in G$. Note that

and for any $h\in H$, $g\in G$, $\mathchar 28953\relax(hg)=\mathchar 28953\relax(hgh^{-1})$, i.e.,

where $a_{h}$ denotes conjugation by $h$. Therefore, by applying Equation (2) to Equation (3) and evaluating the differential of both sides at the identity $e$, it is seen that

i.e., the action of $l_{h}{}_{*}$ on $T_{o}$ corresponds to the action of $\mathop{\rm Ad}\nolimits_{h}$ on ${g}$, which in turn corresponds to the action of $\mathop{\rm Ad}\nolimits_{h}$ on ${g}/{h}$ because ${h}$ is $\mathop{\rm Ad}\nolimits_{H}$-invariant.

Let $G$ be a connected Lie group, $H$ a closed subgroup of $G$, and ${g}$ and ${h}$ the Lie algebras of $G$ and $H$, respectively. The homogeneous space $G/H$ is said to be reductive if there exists a vector subspace ${m}$ of ${g}$ such that ${g}={m}+{h}$ (direct sum), and ${m}$ is $\mathop{\rm Ad}\nolimits_{H}$-invariant, i.e., $\mathop{\rm Ad}\nolimits_{H}({m})\subset{m}$.

For example, the homogeneous space $G/H$ is reductive if $H$ is compact. Our interest in reductive homogeneous spaces lies solely with this class of examples; for others, see ? or ?.

Invariant affine connections

Let $G$ be a Lie transformation group acting on a differentiable manifold $M$. An affine connection $\nabla$ on $M$ is said to be $G$-invariant if for all $g\in G$, $X$, $Y\in{X}(M)$,

Geodesics on $G$ coincide with one-parameter subgroups if and only if $\mathchar 28939\relax(X,X)=0$ for all $X\in{g}$. The classical Cartan-Schouten invariant affine connections on $G$ correspond to $\mathchar 28939\relax(X,Y)\equiv 0$ (the $(-)\hbox{-}$connection), $\mathchar 28939\relax(X,Y)={\mathchoice{{\textstyle{1\over 2}}}{{\textstyle{1\over 2}}}{{\scriptstyle{1\over 2}}}{{\scriptscriptstyle{1\over 2}}}}[X,Y]$ (the $(0)\hbox{-}$connection), and $\mathchar 28939\relax(X,Y)=[X,Y]$ (the $(+)\hbox{-}$connection).

Let $G/H$ be a reductive homogeneous space with a fixed decomposition of the Lie algebra ${g}={m}+{h}$, $\mathop{\rm Ad}\nolimits_{H}({m})\subset{m}$, and $\mathchar 28953\relax\colon G\to G/H$ the natural projection. Any element $X\in{g}$ can be uniquely decomposed into the sum of elements in ${m}$ and ${h}$, which will be denoted by $X_{m}$ and $X_{h}$, respectively. There is a one-to-one correspondence between invariant affine connections on $G/H$ and the set of bilinear functions $\mathchar 28939\relax\colon{m}\times{m}\to{m}$ which are $\mathop{\rm Ad}\nolimits_{H}$-invariant, i.e., $\mathop{\rm Ad}\nolimits_{h}\cdot\mathchar 28939\relax(X,Y)=\mathchar 28939\relax(\mathop{\rm Ad}\nolimits_{h}X,\mathop{\rm Ad}\nolimits_{h}Y)$ for all $X$, $Y\in{m}$, $h\in H$.

Let $t\mapsto\exp tX$ be the one-parameter subgroup generated by $X$ in ${m}$, and denote the curve $t\mapsto\mathchar 28953\relax(\exp tX)$ in $G/H$ by $t\mapsto\mathchar 28941\relax_{X}(t)$. In addition to the requirement that the connection be complete, consider the following conditions on the invariant affine connection on $G/H$.

$(\rm a)$ The curve $\mathchar 28941\relax_{X}$ is a geodesic in $G/H$.

$(\rm b)$ Parallel translation of the tangent vector $Y\in T_{o}$ corresponding to $y\in{m}$ along the curve $\mathchar 28941\relax_{X}$ is given by the differential of $\exp tX$ acting on $G/H$.

Nomizu (?) established the following results concerning invariant affine connections on reductive homogeneous spaces. Recall that the torsion of a connection $\nabla$ on a manifold $M$ is a tensor $T$ of type $(1,2)$ defined by $T(X,Y)=\nabla_{\!X}Y-\nabla_{\!Y}X-[X,Y]$, $X$, $Y\in{X}(M)$. The connection is said to be torsion-free if $T\equiv 0$.

On a reductive homogeneous space $G/H$, there exists a unique invariant connection which is torsion-free and satisfies $(\rm a)$. It is defined by the function $\mathchar 28939\relax(X,Y)={\mathchoice{{\textstyle{1\over 2}}}{{\textstyle{1\over 2}}}{{\scriptstyle{1\over 2}}}{{\scriptscriptstyle{1\over 2}}}}[X,Y]_{m}$ on ${m}\times{m}$.

This connection is called the canonical torsion-free connection on $G/H$ with respect to the fixed decomposition ${g}={m}+{h}$. In the case of a Lie group, it is the Cartan-Schouten $(0)\hbox{-}$connection.

On a reductive homogeneous space $G/H$, there exists a unique invariant connection which satisfies $(\rm b)$. It is defined by the function $\mathchar 28939\relax(X,Y)\equiv 0$ on ${m}\times{m}$.

This connection is called the canonical connection on $G/H$ with respect to the fixed decomposition ${g}={m}+{h}$. In the case of a Lie group, it is the Cartan-Schouten $(-)\hbox{-}$connection.

If $G/H$ is a symmetric homogeneous space, then these two connections coincide. (See ?, ?, or ? for background on symmetric spaces.) From the point of view of the applications we have in mind, the choice of the canonical torsion-free connection on $G/H$ facilitates the computation of geodesics on $G/H$. The choice of the canonical connection on $G/H$ facilitates the computation of parallel translation along curves of the form $t\mapsto\mathchar 28953\relax(\exp tX)$. In the case of a symmetric space, the canonical connection allows both the computation of geodesics and parallel translation along geodesics by conditions $(\rm a)$ and $(\rm b)$ above.

Invariant Riemannian metrics

Let $G$ be a Lie transformation group acting on a differentiable manifold $M$. A tensor field $A$ on $M$ is said to be $G$-invariant if for all $g\in G$, $p\in M$,

In particular, a Riemannian structure $g$ on $M$ is said to be (left) invariant if it is $G$-invariant as a tensor field on $G/H$. That is,

for all $p\in M$, $k\in G$, $X$, $Y\in T_{p}$. In the case of a Lie group $G$, a bi-invariant metric on $G$ is a Riemannian structure on $G$ that is invariant with respect to the left and right action of $G$ on itself.

There may not exist an invariant Riemannian metric on the homogeneous space $G/H$; however, ? provide a proposition that describes the invariant metric structure of all homogeneous spaces considered in this thesis. The following proposition paraphrases Proposition 3.16 of Cheeger and Ebin.

$(1)$ The set of $G$-invariant metrics on $G/H$ is naturally isomorphic to the set of bilinear forms $\langle\,{,}\,\rangle$ on ${g}/{h}\times{g}/{h}$ which are $\mathop{\rm Ad}\nolimits_{H}$-invariant.

$(2)$ If $H$ is connected, the bilinear form $\langle\,{,}\,\rangle$ on ${g}/{h}\times{g}/{h}$ is $\mathop{\rm Ad}\nolimits_{H}$-invariant if and only if for all $\mathchar 28945\relax\in{h}$, $\mathop{\rm ad}\nolimits_{\mathchar 28945\relax}$ is skew-symmetric with respect to $\langle\,{,}\,\rangle$.

$(3)$ If $G$ acts effectively on $G/H$, then $G/H$ admits an invariant metric if and only if the closure of the group $\mathop{\rm Ad}\nolimits_{H}$ in $\mathord{\rm Aut}({g})$, the group of automorphisms of ${g}$, is compact.

$(4)$ If $G$ acts effectively on $G/H$ and $G/H$ is reductive with the fixed decomposition ${g}={m}+{h}$, then there is a one-to-one correspondence between $G$-invariant metrics on $G/H$ and $\mathop{\rm Ad}\nolimits_{H}$-invariant bilinear forms on ${m}\times{m}$. If $G/H$ admits a left invariant metric, then $G$ admits a left invariant metric which is right invariant with respect to $H$; the restriction of this metric to $H$ is bi-invariant.

Setting ${m}={h}^{\perp}$ provides such a decomposition.

$(5)$ If $H$ is connected, then the condition $\mathop{\rm Ad}\nolimits_{H}({m})\subset{m}$ is equivalent to $[{h},{m}]\subset{m}$.

Let $G$ be a Lie group which admits a bi-invariant metric $\langle\,{,}\,\rangle$. Then there is a corresponding left invariant metric, called the normal metric, on the homogeneous space $G/H$ with fixed decomposition ${g}={m}+{h}$, ${m}={h}^{\perp}$, arising from the restriction of $\langle\,{,}\,\rangle$ to ${m}$. For example, let $G$ be a compact semisimple Lie group with Lie algebra ${g}$. The Killing form $\mathchar 28967\relax$ of ${g}$ is negative definite; therefore, $-\mathchar 28967\relax$ naturally defines an invariant Riemannian metric on $G$. The Levi-Civita connection of this metric is the $(0)\hbox{-}$connection of $G$. Let $H$ be a closed subgroup of $G$ such that $G$ acts effectively on $G/H$. Then setting ${m}={h}^{\perp}$ with respect to $-\mathchar 28967\relax$ yields a subspace ${m}$ of ${g}$ such that $\mathop{\rm Ad}\nolimits_{H}({m})\subset{m}$, i.e., $G/H$ is a reductive homogeneous space. Furthermore, $-\mathchar 28967\relax$ restricted to ${m}$ yields an $\mathop{\rm Ad}\nolimits_{H}$-invariant bilinear form on ${m}\times{m}$ and therefore yields a left invariant Riemannian metric on $G/H$. The Levi-Civita connection of this metric is the canonical torsion-free connection on $G/H$.

Formulae for geodesics and parallel translation along geodesics

Let $G$ be a Lie group with the $(0)\hbox{-}$connection, ${g}$ the Lie algebra of $G$, and $X$ a left invariant vector field on $G$ corresponding to $x\in{g}$. Then the unique geodesic in $G$ emanating from $g$ in direction $X_{g}$ is given by the curve $t\mapsto\mathchar 28941\relax(t)$, where

Let $t\mapsto\mathchar 28941\relax_{x}(t)$ be the geodesic in $G$ emanating from the identity $e$ with direction $x\in{g}$, and let $Y(t)$ be the parallel translation of $y\in{g}$ from $e$ to $\exp tX$ along $\mathchar 28941\relax_{X}$. Then

where $e^{x}=\exp x$. For computations involving vector fields on $G$, it is oftentimes convenient to represent a tangent vector $X_{g}$ in $T_{g}G$ ($g\in G$) by a corresponding element $x_{g}$ in ${g}$ defined by the equation $X_{g}=l_{g}{}_{*}x_{g}$. Letting $y(t)\in{g}$ correspond to $Y(t)\in T_{e^{xt}}G$ in this way, it is seen that

Let $G$ be a Lie group with bi-invariant metric $g$ also denoted by $\langle\,{,}\,\rangle$, and $G/H$ a reductive homogeneous space with the normal metric and the fixed decomposition ${g}={m}+{h}$, ${m}={h}^{\perp}$. Denote the natural projection from $G$ onto $G/H$ by $\mathchar 28953\relax$ and let $o=\mathchar 28953\relax(e)$ be the origin in $G/H$. We wish to compute a formula for parallel translation along geodesics in $G/H$. To do this, we view $G$ as principal fiber bundle over $G/H$ with structure group $H$, i.e., we consider the fiber bundle $G(G/H,H)$ with its canonical torsion-free connection [KobayashiandNomizu, Chap. 1, § 5; Chap. 2].

For every element $x\in{m}$, there is a unique element $X_{o}\in T_{o}(G/H)$ given by Equation (4). For $t$ small enough, define the vector field $X$ along the geodesic $t\mapsto\exp tx\cdot o$ in $G/H$ by setting $X_{e^{xt}}=l_{e^{xt}}{}_{*}X_{o}$. There is a unique horizontal lift $\bar{X}\in{X}(G)$ of a smooth extension of $X$. Let $Y$ be a parallel vector field along the geodesic $t\mapsto\exp(tx)\cdot o$ and denote $Y$ evaluated at the point $\exp(tx)\cdot o$ by $Y(t)$. For each $t\in{\bf R}$, define $Y_{o}(t)\in T_{o}(G/H)$ and $y(t)\in{m}$ by the equation

By the definition of the Levi-Civita connection, we have

The vector field $Y$ is parallel along the geodesic; therefore,

by definition. Computing the rightmost term of Equation (8), we find that

Combining Equations (8), (9), (10), and (11), we have proved:

Let $M=G/H$ be a reductive homogeneous space which admits an invariant metric $\langle\,{,}\,\rangle$ and which has the fixed decomposition ${g}={m}+{h}$, ${m}={h}^{\perp}$. Denote the origin of $M$ by $o=\mathchar 28953\relax(e)$, where $\mathchar 28953\relax\colon G\to M$ is the natural projection. Let $x$ and $y_{0}$ be vectors in ${m}$ corresponding to the tangent vectors $X$ and $Y_{0}$ in $T_{o}(G/H)$. The parallel translation $Y(t)$ of $Y_{0}$ along the geodesic $t\mapsto\exp_{o}tX=e^{xt}\cdot o$ in $M$ is given in the following way. Define $Y_{o}(t)\in T_{o}(G/H)$ by the equation $Y(t)=l_{e^{xt}}{}_{*}Y_{o}(t)$, and let $y(t)\in{m}$ correspond to $Y_{o}(t)\in T_{o}(G/H)$. The vector $y(t)$ satisfies the ordinary linear differential equation

In the case of a Lie group $M=G$, we may take ${m}={g}$; thus Equation (12) reduces to

whose solution is given by Equation (7). In the case where $G/H$ is a symmetric space, the decomposition ${g}={m}+{h}$ satisfies the properties

Therefore $\dot{y}\equiv 0$ because the ${m}$-component of $[x,y]$ vanishes. Thus $y(t)\equiv y_{0}$.

Examples

The general linear group $\mathord{\elvbit G\!L}({n})$ is the set of all real $n$-by-$n$ invertible matrices. $\mathord{\elvbit G\!L}({n})$ is easily verified to be a Lie group of dimension $n^{2}$ whose Lie algebra $\mathord{gl}({n})$ is the vector space of all $n$-by-$n$ matrices with bracket $[X,Y]=XY-YX$. The orthogonal group $\mathord{\elvib O}({n})$ is the subgroup of $\mathord{\elvbit G\!L}({n})$ given by

It is well-known that $\mathord{\elvib O}({n})$ is a Lie group of dimension $n(n-1)/2$ with two connected components. The identity component of $\mathord{\elvib O}({n})$ is called the special orthogonal group $\mathord{\elvbit S\!O}({n})$, which is defined by

The Lie algebra of $\mathord{\elvbit S\!O}({n})$, denoted by $\mathord{so}({n})$, is the set of all $n$-by-$n$ skew-symmetric matrices, i.e.,

The orthogonal group is the group of all isometries of the vector space ${\bf R}^{n}$ (endowed with the standard inner product $\langle x,y\rangle=x^{\scriptscriptstyle\rm T}y=\sum_{i}x^{i}y^{i}$) which fix the origin. The special orthogonal group is the group of all orientation preserving isometries of ${\bf R}^{n}$ which fix the origin.

The orthogonal group $\mathord{\elvib O}({n})$ is compact, and therefore admits a bi-invariant metric, which is given by the negative of the Killing form of $\mathord{so}({n})$. A computation shows that for $X$, $Y\in\mathord{so}({n})$,

where the first trace is the trace of endomorphisms of $\mathord{so}({n})$, and the second trace is the trace of $n$-by-$n$ matrices. In case $n=2$, take the bilinear form $(X,Y)\mapsto\mathop{\rm tr}\nolimits XY$. Furthermore, $\mathord{\elvib O}({n})$ is semisimple, therefore $-\mathchar 28967\relax$ is positive definite and thus defines a bi-invariant metric on $\mathord{\elvib O}({n})$. This is the natural bi-invariant metric on $\mathord{\elvib O}({n})$.