Kosmann lift

In differential geometry, the Kosmann lift,^[1]^[2] named after Yvette Kosmann-Schwarzbach, of a vector field $X\,$ on a Riemannian manifold $(M,g)\,$ is the canonical projection $X_{K}\,$ on the orthonormal frame bundle of its natural lift $\hat{X}\,$ defined on the bundle of linear frames.^[3]

Generalisations exist for any given reductive G-structure.

1 Introduction
2 Definition
3 See also
4 Notes
5 References

Introduction

In general, given a subbundle $Q\subset E\,$ of a fiber bundle $\pi_{E}\colon E\to M\,$ over $M$ and a vector field $Z\,$ on $E$ , its restriction $Z\vert_Q\,$ to $Q$ is a vector field "along" $Q$ not on (i.e., tangent to) $Q$ . If one denotes by $i_{Q} \colon Q\hookrightarrow E$ the canonical embedding, then $Z\vert_Q\,$ is a section of the pullback bundle $i^{\ast}_{Q}(TE) \to Q\,$ , where

$i^{\ast}_{Q}(TE) = \{(q,v) \in Q \times TE \mid i(q) = \tau_{E}(v)\}\subset Q\times TE,\,$

and $\tau_{E}\colon TE\to E\,$ is the tangent bundle of the fiber bundle $E$ . Let us assume that we are given a Kosmann decomposition of the pullback bundle $i^{\ast}_{Q}(TE) \to Q\,$ , such that

$i^{\ast}_{Q}(TE) = TQ\oplus \mathcal M(Q),\,$

i.e., at each $q\in Q$ one has $T_qE=T_qQ\oplus \mathcal M_u\,,$ where $\mathcal M_{u}$ is a vector subspace of $T_qE\,$ and we assume $\mathcal M(Q)\to Q\,$ to be a vector bundle over $Q$ , called the transversal bundle of the Kosmann decomposition. It follows that the restriction $Z\vert_Q\,$ to $Q$ splits into a tangent vector field $Z_K\,$ on $Q$ and a transverse vector field $Z_G,\,$ being a section of the vector bundle $\mathcal M(Q)\to Q.\,$

Definition

Let $\mathrm F_{SO}(M)\to M$ be the oriented orthonormal frame bundle of an oriented $n$ -dimensional Riemannian manifold $M$ with given metric $g\,$ . This is a principal ${\mathrm S\mathrm O}(n)\,$ -subbundle of $\mathrm FM\,$ , the tangent frame bundle of linear frames over $M$ with structure group ${\mathrm G\mathrm L}(n,\mathbb R)\,$ . By definition, one may say that we are given with a classical reductive ${\mathrm S\mathrm O}(n)\,$ -structure. The special orthogonal group ${\mathrm S\mathrm O}(n)\,$ is a reductive Lie subgroup of ${\mathrm G\mathrm L}(n,\mathbb R)\,$ . In fact, there exists a direct sum decomposition $\mathfrak{gl}(n)=\mathfrak{so}(n)\oplus \mathfrak{m}\,$ , where $\mathfrak{gl}(n)\,$ is the Lie algebra of ${\mathrm G\mathrm L}(n,\mathbb R)\,$ , $\mathfrak{so}(n)\,$ is the Lie algebra of ${\mathrm S\mathrm O}(n)\,$ , and $\mathfrak{m}\,$ is the $\mathrm{Ad}_{\mathrm S\mathrm O}\,$ -invariant vector subspace of symmetric matrices, i.e. $\mathrm{Ad}_{a}\mathfrak{m}\subset\mathfrak{m}\,$ for all $a\in{\mathrm S\mathrm O}(n)\,.$

Let $i_{\mathrm F_{SO}(M)} \colon \mathrm F_{SO}(M)\hookrightarrow \mathrm FM$ be the canonical embedding.

One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle $i^{\ast}_{\mathrm F_{SO}(M)}(T\mathrm FM) \to \mathrm F_{SO}(M)$ such that

$i^{\ast}_{\mathrm F_{SO}(M)}(T\mathrm FM)=T\mathrm F_{SO}(M)\oplus \mathcal M(\mathrm F_{SO}(M))\,,$

i.e., at each $u\in \mathrm F_{SO}(M)$ one has $T_u\mathrm FM=T_u \mathrm F_{SO}(M)\oplus \mathcal M_u\,,$ $\mathcal M_{u}$ being the fiber over $u$ of the subbundle $\mathcal M(\mathrm F_{SO}(M))\to \mathrm F_{SO}(M)$ of $i^{\ast}_{\mathrm F_{SO}(M)}(V\mathrm FM) \to \mathrm F_{SO}(M)$ . Here, $V\mathrm FM\,$ is the vertical subbundle of $T\mathrm FM\,$ and at each $u\in \mathrm F_{SO}(M)$ the fiber $\mathcal M_{u}$ is isomorphic to the vector space of symmetric matrices $\mathfrak{m}$ .

From the above canonical and equivariant decomposition, it follows that the restriction $Z\vert_{\mathrm F_{SO}(M)}$ of an ${\mathrm G\mathrm L}(n,\mathbb R)$ -invariant vector field $Z\,$ on $\mathrm FM$ to $\mathrm F_{SO}(M)$ splits into a ${\mathrm S\mathrm O}(n)$ -invariant vector field $Z_{K}\,$ on $\mathrm F_{SO}(M)$ , called the Kosmann vector field associated with $Z\,$ , and a transverse vector field $Z_{G}\,$ .

In particular, for a generic vector field $X\,$ on the base manifold $(M,g)\,$ , it follows that the restriction $\hat{X}\vert_{\mathrm F_{SO}(M)}\,$ to $\mathrm F_{SO}(M)\to M$ of its natural lift $\hat{X}\,$ onto $\mathrm FM\to M$ splits into a ${\mathrm S\mathrm O}(n)$ -invariant vector field $X_{K}\,$ on $\mathrm F_{SO}(M)$ , called the Kosmann lift of $X\,$ , and a transverse vector field $X_{G}\,$ called the von Göden lift of $X\, .$

Notes

^ Fatibene L., Ferraris M., Francaviglia M. and Godina M. (1996), A geometric definition of Lie derivative for Spinor Fields, in: Proceedings of the 6th International Conference on Differential Geometry and Applications, August 28th– September 1st 1995 (Brno, Czech Republic), Janyska J., Kolář I. & J. Slovák J. (Eds.), Masaryk University, Brno, pp. 549–558
^ Godina M. and Matteucci P. (2003), Reductive G-structures and Lie derivatives, Journal of Geometry and Physics 47, 66–86
^ Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1, Wiley-Interscience, ISBN 0470496479 (Example 5.2) pp. 55-56

References

Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley-Interscience, ISBN 0471157333
Kolář, Ivan; Michor, Peter; Slovák, Jan (1993) (PDF), Natural operators in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf
Sternberg, S. (1983), Lectures on Differential Geometry (2nd ed.), New York: Chelsea Publishing Co., ISBN 0-8218-1385-4
Fatibene, Lorenzo; Francaviglia, Mauro (2003), Natural and Gauge Natural Formalism for Classical Field Theories, Kluwer Academic Publishers, ISBN 978-1402017032

Kosmann lift

Contents

Introduction

Definition

See also

Notes

References