[14] is an extension of the smooth preparation theorem of Malgrange to Banach spaces.
Starting from an idea by Eells I tried to lay good foundations for the theory of manifolds of smooth mappings between finite dimensional manifolds; this was done before only for compact source manifolds. In the beginning I used the calculus $C^\infty_c$ in the notation of Keller, Springer LN 417. Paper [9] develops the foundations, [12] treats the diffeomorphism group for a non-compact manifold, [13] treats the principal bundle of embeddings with structure group the diffeomorphism group; these papers culminate in the monography [C]. [16] continues this theory: Here one finds the following results for manifolds modeled on nuclear (LF)-spaces: smooth partitions of unity, continuous derivations of the algebra of smooth functions are exactly the tangent vectors, and the De Rham cohomology equal the singular cohomology with real coefficients.
Then Alfred Fr\ölicher and Andreas Kriegl developed their calculus in infinite dimensions (I was the supervisor of Kriegl's thesis) which made much easier most of the developments. Papers [17] and [18] develop a cartesian closed theory of infinite dimensional manifolds, where (necessarily) there are no longer charts; all is based on the structure of smooth curves. This is a successful development, but it is so complicated that it was never used by anyone.
For manifolds of mappings the Frölicher-Kriegl calculus is applied in the following papers: [19] is a review on applying the hard Nash-Moser implicit function theorem. [24] states that the (singular) cohomology of a diffeomorphism group is the continuous Lie algebra cohomology (Gelfand-Fuks) of the Lie algebra of all vector fields with values in the module of all smooth functions on the diffeomorphism group. [28] shows, that $b$-incompleteness (a possible notion for space times) is stable. [21] and [30] gives a smooth account of the automorphism group of a principal bundle and its action on the space of connections; this was already well known in the Sobolev approach.
[36] gives a concise description of the space of smooth vectors of a unitary representation of a Lie group and shows that the group acts smoothly on it, may be differentiated, and gives a rigorous setting to infinitesimal generators for infinite dimensional representations. Every unitary representation admits a moment mapping: this was clear before in a not rigorous way, or with a lot more effort.
[40] studies the action of the diffeomorphism group on the space of immersions. Where the action is free, this turns out to be a principal bundle; in general we could show the existence of slices and start a description of the orbit space.
[39] investigates the natural Riemannian geometry on the space of all Riemannian metrics, solves the geodesic equation explicitly, shows that the exponential mapping is a diffeomorphism on its image, and gives all Jacobi fields explicitly. This work takes up once more ideas by Ebin, DeWitt, Freed and Groisser, and puts it in a rigorous setting. [43] extends this to the space of all bilinear structures on a manifold, show that metrics of a fixed signature and weakly symplectic structures are geodesically closed submanifolds, and describes a useful splitting of the manifold of all metrics. [52] treats the infinite dimensional pseudo-Riemannian manifold of all almost hermitian structures, computes the geodesic equation etc. explicitly, and describes many first integrals of the geodesic equation. Unfortunately there is no explicit solution of the geodesic equation.
[33] and [47] show that under very general assumptions even on infinite dimensional manifolds all bounded multiplicative linear functionals on the algebra of smooth functions are just the point evaluations (Milnor's and Stasheff's exercise). This was the starting point of a series of papers by several groups of authors. My ultimate aim in this activity is to generalize the Theory of Weil-functors (see [23]) to infinite dimensions. The next step in this direction is taken in [67] containing the beginnings of the theory of product preserving functors in infinite dimensions.
[22] studies $C^\infty$-algebras. These were introduced by F. X. Lawvere as commutative algebras where one can evaluate (canonically) smooth functions (not only polynomials). We give a description in term of functional analysis by equipping each such algebra with the direct limit locally convex topology with respect to the nuclear Fréechet spaces of smooth functions on $R^n$, and we found a condition for a locally m-convex commutative algebra to be a $C^\infty$-algebra (\v Cebyshev condition). A corresponding theory of non-commutative $C^\infty$-algebras would be very helpful ([51] contains an example). [59] characterizes algebras of smooth functions on finite dimensional manifolds among all $C^\infty$-algebras.
The paper [62] takes up the notion of an infinite dimensional regular Lie group in the sense of Omori et. al. , as generalized by Milnor in his Les Houches lecture Notes on infinite dimensional Lie groups. It is a systematic study of this notion and the main results are: regular Lie groups are stable under constructions like semidirect products and extensions, which paves the way for the computation of the derivative of the evolution operator. This in turn is then used to show, that a principal bundle with a regular Lie group as structure group admits parallel transports for each connection, and that flat connections are integrable in a very strong sense. In the beginning examples show that solving ordinary differential equations in infinite dimensions is a very non-trivial business. [67] contains the beginnings of the theory of product preserving functors in infinite dimensions, as a (nontrivial) generalization of [23].
[38] is an account of some applications of the Fröicher-Kriegl calculus in, including also the surprising fact that on Hilbert spaces there are continuous derivations on the algebra of smooth functions which are not usual tangent vectors: they involve higher derivatives. Also direct limit manifolds like $O(\infty,R)$, Grassmanian, and so on, are identified as being real analytic (or even holomorphic) manifolds and Lie groups, where even the exponential mapping is a local diffeomorphism.
The monograph ( [G]: The Convenient Setting of Global Analysis (Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. 618 pages) is a thorough introduction into smooth, holomorphic, and real analytic analysis in infinite dimensions along the Frölicher-Kriegl line from the point of view of functional analysis. Then it treats foundational questions of differential geometry in infinite dimensions like existence of partitions of unity, several kinds of tangent vectors and differential forms, and cohomology. The last third of the book is devoted to applications: regular infinite dimensional Lie groups, the key results of manifolds of mappings and of direct limit manifolds, infinite dimensional symplectic manifolds, and perturbation of unbounded operators on Hilbert spaces.
Some results on Real analytic mappings : It was stated by Leslie with a wrong proof that the group of real analytic diffeomorphisms of a compact manifold is a smooth Lie group. In [37] we develop `the' cartesian closed theory of real analytic mappings between convenient locally convex spaces. This is much more subtle than to require local convergence of the Taylor series and involve quite deep theorems of analysis and functional analysis. The resulting theory is easy to understand and to apply and leads to the desired result that the group of real analytic diffeomorphisms of a compact manifold is indeed a real analytic Lie group, and opens the way to apply the real analytic theory in the general theory of manifolds of mappings.
In [95] we show that continuous and smooth homotopy groups agree on infinite dimensional convenient manifolds, where one cannot find convex charts in general.
[26] determines all natural concomitants between pairs of vector valued differential forms, a 10-dimensional vector space in general, where the most interesting object is the Frölicher-Nijenhuis bracket. [27] shows that up to a constant multiple the Schouten-Nijenhuis bracket is the unique natural concomitant between multi vector fields.
The monograph ( [F] : Natural operations in differential geometry) arose from the Middle-European seminar on differential geometry (Brno -- Vienna) and is a thorough account of all kinds and uses of naturality in differential geometry. It might the bible of this field for some time to come.
[20] showed that for symplectic manifolds there is a unique extension of the Poisson bracket for smooth function to a Z-graded Lie bracket for differential forms and a an extension of the Hamiltonian vector field mapping to a mappings of space of differential forms to the space of vector valued differential forms which is then a homomorphism where the image space carries the Frölicher-Nijenhuis bracket. This paper gave rise to many articles, mainly in Mathematical Physics. Recently Janusz Grabowski has solved completely the problem attacked in [20], namely he has extended the Poisson bracket from functions to a graded Lie bracket on the space of all differential forms, on a Poisson manifold, which is, however, bilinear differential operator of degree 2.
[32] shows that the F-N bracket can lead to a theory of connections, curvature and cocurvature, and Bianchi identity in a very general context. In the context of fiber bundles this is explained in [29] and more fully developed in ([E]: Gauge theory for fiber bundles): it is shown that nearly all the gauge theory of principal bundles can be carried over to fiber bundles without structure group: connections, curvature, parallel transport, holonomy group and algebra; a vast generalization of the Ambrose-Singer theorem shows that if the holonomy Lie algebra is finite dimensional then the bundle and the connection are actually induced by principal ones. Certain types of characteristic classes (coming from equivariant of the Lie algebra of all vector fields; there are no invariants!) are constructed and put into relation with algebraic topology. There are many open ends and I will pursue this line of research also in the future.
[45] uses this theory to show that each `strong' system in the sense of Modugno which the property that all vertical vector fields of the system are complete, is actually associated to a principal bundle. Modugno has invented his kind of systems in order to be able to write field equations without group theory. My result now shows that group theory come in by the back door again via the holonomy group of any connection of the system and one is back to the starting point.
[34] computed the cohomology of certain natural differential on the space of vector valued differential forms which arose in the thesis of Schicketanz as the product of the De Rham cohomology with the space of tracefree differential forms.
In [54] for any manifold M the following result is shown: There is a canonical mapping from the space of sections of the bundle $\Lambda T^*M\otimes STM$ to $\Omega(T^*M;T(T^*M))$. It is shown that this is a homomorphism on $\Omega(M;TM)$ for the Frölicher-Nijenhuis brackets, and also on $\Gamma(STM)$ for the Schouten bracket of symmetric multivector fields. But the whole image is not a subalgebra for the Frölicher-Nijenhuis bracket on $\Omega(T^*M;T(T^*M))$.
[68] introduces the Jacobi flow on the second tangent bundle of a (say Riemannian) manifold, given by a vector field, whose flow lines have various projections onto a geodesic in $M$, its velocity field in $TM$, and a Jacobi field along the geodesic.
In [72] it is shown that on the space of generalized connections on a fiber bundle (where there is no finite dimensional Lie group acting as a structure group) does not admit slices in any sense for the action of the gauge group (which is the the group of fiber preserving diffeomorphisms). This is in contrast to the case of $G$-bundles, where there is a finite dimensional Lie group acting as a structure group, and where the correponding slice theorem is at the basis of Donaldsons striking applications of Yang-Mills theory to 4-dimensional topology.
[31] starts from the formula for the multiplication of quaternions $(X,r).(Y,s)= ([X,Y]+rY+sX, rs+B(X,Y))$ where $X,Y\in \mathfrak{so}(3,R)$ are in the Lie algebra of $SO(3,R)$ and $B=-\langle\quad,\quad\rangle$ is its Cartan Killing form. We investigate for which Lie algebras this gives an associative multiplication (nilpotent of order 2, $\mathfrak{so}(3,R)$, $\mathfrak{sl}(2,\Bbb R)$), and we characterize algebras given by this construction by a `Clifford-trace' property.
[35]: If a Lie algebra is the direct sum of two subalgebras, then it may be written as `derivatively knitted product' of the two. The group version of this is well known under the name `Zappa-Szep' product. The decomposition for representations is also given. This arose from formulas intertwining the Frölicher-Nijenhuis and the Nijenhuis-Richardson brackets.
[42]: The Nijenhuis-Richardson bracket arose in differential geometry, but in the 60's one found out that it recognizes Lie brackets and gives a formula for its Chevalley-cohomology. It is very useful in studying deformations of Lie algebra structures. In this paper we found a series of brackets: The (n+1)-graded bracket recognizes n-graded Lie brackets, their modules, and gives the right formula for the Chevalley cohomology. This also used to study deformations. This method is taken up taken up in [66] as a convenient means to find the (some) definition of associative or Lie structures with more than 2 entries. This has been become popular recently in the context of Nambu mechanics.
The papers [46], [48], [61], [55], and [63] have also a strong algebraic flavor and many results which belong to the theory of non-commutative rings and their modules. For a detailed description of their content see the section on non-commutative geometry below.
[44] shows that $1/k!$ the $k$-th derivative of group commutator expression of curves of local flows of vector fields equals the corresponding Lie bracket expression of the generating vector fields, and generalizations thereof. For two vector fields this is a well known tool in differential geometry and control theory.
[51] constructs characteristic classes for $G$-structures, where one also inserts the displacement form (or the soldering form on vector bundles) into the invariant polynomials. If the $G$-structure is 1-integrable, characteristic classes result. There are some examples of new classes given. In [57] the same approach is tried for Cartan connections: but here no new classes emerge, they are just classes of an appropriate bundle with enlarged structure group.
In [58] for Riemannian $G$-actions with sections the space of `horizontal' $G$-invariant differential forms is shown to be isomorphic to the space of differential forms on the section which invariant under the Weyl group, under some condition, which is then removed completely in [64]. This result was carried over to reductive group actions on affine varieties by [Brion, Michel: Differential forms on quotients by reductive group actions. Proc. Amer. Math. Soc.]
In [56] the idea is to study just an infinitesimal action of a finite dimensional Lie algebra, where the vector fields are not complete, so that it cannot be integrated to an action of a Lie group. Here we introduce principal connections, curvature, parallel transport, and characteristic classes.
[50] views Radon transforms $R:C^\infty_c(M)\to C^\infty(\Sigma;)$ as embeddings $\Sigma;\to D'(M)$ into the space of distributions and computes a kind of second fundamental form of the submanifold $\Sigma;$ of the space of distributions $ D'(M)$. This gives quite surprising formulas.
[53] treats Poisson structures and their reductions in the following way: On a cotangent bundle $T^*G$ of a Lie group $G$ one can describe the standard Liouville form $\Theta;$ and the symplectic form $d\Theta;$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on itself), and also in terms of the left Maurer-Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic structures on $T^*G$ and to Poisson structures by replacing the canonical momenta of the right or left actions of $G$ on itself by arbitrary ones, followed by reduction (to $G$ cross a Weyl-chamber, e. g. ). This method also works on principal bundles. A generalization of this setting is interpreted in [71] as a generalization of the notion of an integrable system. In [70] Poisson structures on Lie groups are studied in detail. First the notion of a Lie bialgebra or Lie Poisson structure is investigated and reviewed and spelled out in terms of Gauss triples which are a generalization of Manin pairs. This is then carried over to Lie groups, in particular the double group and its various Poisson structures and dressing transformations are studied in detail.
The paper [65] belongs to analysis, but it originated in questions which arose in the study of invariants of Lie group actions. Assume that we have a polynomial whose coefficients depends smoothly on a real parameter. Can one choose the roots smoothly in that parameter. This is answered completely in the case that all roots are real: yes if no two roots meet of inifinite order, no in general, but the roots can always be chosen differentiable, but not $C^1$ for degree $\ge 3$, and in degree 2 can always be chosen twice differentiable but not $C^2$. Also complex roots are studied, and applications to perturbation theory of operators are given. In particular there is a result which is genuinly stronger than any in the book of Kato on perturbation theory of operators which allows under some assumptions to choose the eigenvalues of a smooth 1-parameter family of selfadjoint operators with compact resolvent in a smooth way. The original question can be restated as follows: The permutation group acts by coordinate permutations on $\Bbb R^n$ and the elementary symmetric polynomials $\sigma_i$ (the coefficients of the polynomial whose roots are the coordinates of $\Bbb R^n$) are generators for the ring of all invariants polynomials. Can one lift smooth curves in this setting over $\sigma=(\sigma_1,\dots,\sigma_h): \Bbb R^n \to \Bbb R^n$? In [73] is is shown that the answer is yes for an arbitrary linear representation of a compact Lie group, under similar conditions as in the case of polynomials.
A counterexample in [65] turned out to be wrong by a computational error. This is taken up in [91], where the results of [65] are strenghened to show that one can choose the roots a $C^{3n}$-curve of polynomials with only real roots always twice differentiable, but not better. In [89] this is used to derive that the eigenvalues of a smooth curve of unbounded selfadjoint operators with compact resolvent on Hilbert space can always be chosen twice differentiable, and $C^1$ if the curve is only $C^{1,\alpha}$ Hoelder differentiable. In a certain sense this results are best possible.
In the paper [74] the known upper bound $2k-1$ on the multiplicity of the k-th eigenvalue of the Laplace operator with Dirichlet boundary condition on any domain with smooth boundary in the plane is improved to $2k-3$, for $k>2$.
In the paper [77] a class of $n$-ary Poisson structures of constant rank is indicated, and it is proved that the ternary Poisson brackets are exactly those which are defined by a decomposable $3$-vector field. The key point is the proof of a lemma which tells that an $n$-vector $(n\geq3)$ is decomposable iff all its contractions with up to $n-2$ covectors are decomposable.
In the paper [83] a result of Palais is reproved: For any manifold with vector field there is a universal completion to a manifold with complete vectorfield which, however, can be a non-Hausdorff manifold. In [92] this is applied to vector fields on infinite dimensional manifolds, namely to the Burgers' equation (a nonlinear PDE) to identify its universal completion: One can prolong the solutions beyond the shocks. In the paper [96] this is extended to completions of actions of Lie algebras on manifolds to actions of Lie groups. But here worse can happen than just non-Hausdorff manifolds.
In the paper [88] it is characterized when one can lift tensor fields (even with poles) from the orbit space of a finite group action on a complex representation space to the representation space. This is used to lift automorphisms of the orbifold to the representations space. The paper [90] is the algebraic geometry companion of the former paper, where similar results are shown for algebraic actions of finite groups and affine varieties. The lift of automorphisms is the first result of extending field automorphisms from a field to a finite extension field.
In [87] we treat the Cayley mapping from an algebraic group to its Lie algebra which is induced by a representation: For the spin representation one obtains the classical Cayley transform $A\mapsto (1-A)/(1+A)$ for matrices, multiplied with an algebraic function which kills the polar divisor. In this paper we proved also a real inequality which was later pushed to its limit in `A Class of Exponential Inequalities', by Jonathan Borwein and Roland Girgensohn, to appear in `Mathematical Inequalities and Applications'.
The paper [93] belongs to invariant theory, and it shows that generalized polarizations can be used to descrine all invariants on direct sums of representations in term of the original invariants, in some cases. I thus solves one of the open problems in invariant theory.
The paper [94] extends the attemps to lift curves over invariants to general mappings: in general there is no paositive answer, but we give a sufficient condition for lifting.
The paper [85] studies the projections of geodesics on a Riemannian manifold in the orbit space of a proper isometric group action: Geodesics which are orthogonal to orbits go to local length minimizing curves, whereas other geodesics in some examples go to highly interesting curves like generalizations of the Calogero-Moser intergrable system which exists on each polar representation of a compact Lie group.
The paper [97] studies reflection groups on general Riemannian manifolds: chambers, angles between walls, reconstruction of the manifold from the chamber, relation to the fundamental group, etc.