The homotopy type of the space of degree 0 immersed plane curves.
The space B = Imm (S, R) / Diff (S) of all immersions of rotation degree 0 in the plane modulo reparameterizations has homotopy groups π(B ) = Z, π(B ) = Z, and π(B ) = 0 for k ≥ 3.
Riemannian geometries on spaces of plane curves
We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from to the plane modulo the group of diffeomorphisms of , acting as reparametrizations. In particular we investigate the metric, for a constant , where is the curvature of the curve and , are normal vector fields to . The term is a sort of geometric Tikhonov regularization because, for , the geodesic distance between any two distinct curves is 0, while for the...
Natural transformations in differential geometry
Characterizing algebras of -functions on manifolds
Among all -algebras we characterize those which are algebras of -functions on second countable Hausdorff -manifolds.
Product preserving functors of infinite-dimensional manifolds
The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of -algebras.
Closed surfaces with different shapes that are indistinguishable by the SRNF
The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [5], provides a way of representing immersed surfaces in , and equipping the set of these immersions with a “distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of . Thus, it induces a distance function on the shape space of immersions, i.e., the space...
Commutators of flows and fields
The well known formula for vector fields , is generalized to arbitrary bracket expressions and arbitrary curves of local diffeomorphisms.
Homology and modular classes of Lie algebroids
For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evens, J.-H. Lu, and A. Weinstein.
Remarks on the Schouten-Nijenhuis bracket
Knit products of graded Lie algebras and groups
Let and be graded Lie algebras whose grading is in or , but only one of them. Suppose that is a derivatively knitted pair of representations for , i.e. and satisfy equations which look “derivatively knitted"; then , endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra . This graded Lie algebra is called the knit product of and . The author investigates the general situation for any graded Lie subalgebras and of a graded...
The cohomology of the diffeomorphism group of a manifold is a Gelfand-Fuks cohomology
All natural concomitants of vector valued differential forms
A note on n-ary Poisson brackets
An -ary Poisson bracket (or generalized Poisson bracket) on the manifold is a skew-symmetric -linear bracket of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order , i.e., being the symmetric group. The notion of generalized Poisson bracket was introduced by et al. in [J. Phys. A, Math. Gen. 29, No. 7, L151–L157 (1996; Zbl 0912.53019) and J. Phys. A, Math. Gen. 30, No. 18, L607–L616 (1997; Zbl 0932.37056)]. They established...
The flow completion of a manifold with vector field.
Regular infinite dimensional Lie groups.
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms.
Completing Lie algebra actions to Lie group actions.
On a construction connecting Lie algebras with general algebras
Tensor fields and connections on holomorphic orbit spaces of finite groups.
Extensions of super Lie algebras.
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