### Remarks on the Schouten-Nijenhuis bracket

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Let $A={\u2a01}_{k}{A}_{k}$ and $B={\u2a01}_{k}{B}_{k}$ be graded Lie algebras whose grading is in $\mathcal{Z}$ or ${\mathcal{Z}}_{2}$, but only one of them. Suppose that $(\alpha ,\beta )$ is a derivatively knitted pair of representations for $(A,B)$, i.e. $\alpha $ and $\beta $ satisfy equations which look “derivatively knitted"; then $A\oplus B:={\u2a01}_{k,l}({A}_{k}\oplus {B}_{l})$, endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A{\oplus}_{(\alpha ,\beta )}B$. This graded Lie algebra is called the knit product of $A$ and $B$. The author investigates the general situation for any graded Lie subalgebras $A$ and $B$ of a graded...

An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\{,\cdots ,\}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$, i.e., $$\sum _{\sigma \in {S}_{2n-1}}(sign\sigma )\{\{{f}_{{\sigma}_{1}},\cdots ,{f}_{{\sigma}_{n}}\},{f}_{{\sigma}_{n+1}},\cdots ,{f}_{{\sigma}_{2n-1}}\}=0,$$ ${S}_{2n-1}$ 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 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.

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from ${S}^{1}$ to the plane modulo the group of diffeomorphisms of ${S}^{1}$, acting as reparametrizations. In particular we investigate the metric, for a constant $A>0$, ${G}_{c}^{A}(h,k):={\int}_{{S}^{1}}(1+A{\kappa}_{c}{\left(\theta \right)}^{2})\langle h\left(\theta \right),k\left(\theta \right)\rangle \left|{c}^{\text{'}}\left(\theta \right)\right|d\theta $ where ${\kappa}_{c}$ is the curvature of the curve $c$ and $h$, $k$ are normal vector fields to $c$. The term $A{\kappa}^{2}$ is a sort of geometric Tikhonov regularization because, for $A=0$, the geodesic distance between any two distinct curves is 0, while for $A>0$ the...

The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of ${C}^{\infty}$-algebras.

Among all ${C}^{\infty}$-algebras we characterize those which are algebras of ${C}^{\infty}$-functions on second countable Hausdorff ${C}^{\infty}$-manifolds.

The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [5], provides a way of representing immersed surfaces in ${\mathbb{R}}^{3}$, 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 ${\mathbb{R}}^{3}$. Thus, it induces a distance function on the shape space of immersions, i.e., the space...

The well known formula $[X,Y]={\textstyle \frac{1}{2}}{\textstyle \frac{{\partial}^{2}}{\partial {t}^{2}}}{|}_{0}{(}_{-t}^{Y}{\xf8}_{-t}^{X}{\xf8}_{t}^{Y}{\xf8}_{t}^{X})$ for vector fields $X$, $Y$ is generalized to arbitrary bracket expressions and arbitrary curves of local diffeomorphisms.

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