Kosmann lift explained

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.

Introduction

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 , wherei ^{\ast} _{Q} (TE) = \{(q,v) \in Q x TE \mid i(q) = \tau _{E} (v)\}\subset Q x 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 thati ^{\ast} _{Q} (TE) = TQ ⊕ l M(Q), i.e., at each q\in Q one has T _{qE=T} _{qQ ⊕} l M _{u,} where l M _{u} is a vector subspace of T _{qE} and we assume l 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 l M(Q)\to Q.

Definition

Let

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 { S O}(n) -subbundle of FM , the tangent frame bundle of linear frames over M with structure group { G L}(n, R) .By definition, one may say that we are given with a classical reductive { S O}(n) -structure. The special orthogonal group { S O}(n) is a reductive Lie subgroup of { G L}(n, R) . In fact, there exists a direct sum decomposition ak{gl}(n)= ak{so}(n) ⊕ ak{m} , where ak{gl}(n) is the Lie algebra of { G L}(n, R) , ak{so}(n) is the Lie algebra of { S O}(n) , and ak{m} is the Ad _{SO} -invariant vector subspace of symmetric matrices, i.e. Ad _{a} ak{m}\subset ak{m} for all a\in{ S O}(n).

Let

i _{F _{SO} (M)} \colon F _{SO} (M)\hookrightarrow FM be the canonical embedding. One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle i ^{\ast} _{F _{SO} (M)} (T FM) \to F _{SO} (M) such thati ^{\ast} _{F _{SO} (M)} (T FM)=T F _{SO} (M) ⊕ l M( F _{SO} (M)),

i.e., at each

u\in F _{SO} (M) one has T _{uFM=T} _{u} F _{SO} (M) ⊕ l M _{u,} l M _{u} being the fiber over u of the subbundle l M( F _{SO} (M))\to F _{SO} (M) of i ^{\ast} _{F _{SO} (M)} (V FM) \to F _{SO} (M) . Here, V FM is the vertical subbundle of T FM and at each u\in F _{SO} (M) the fiber l M _{u} is isomorphic to the vector space of symmetric matrices ak{m} .

From the above canonical and equivariant decomposition, it follows that the restriction

Z\vert _{F _{SO} (M)} of an { G L}(n, R) -invariant vector field Z on FM to F _{SO} (M) splits into a { S O}(n) -invariant vector field Z _{K} on F _{SO} (M) , called the Kosmann vector field associated with Z , and a transverse vector field Z _{G} .X on the base manifold (M,g) , it follows that the restriction \hat{X}\vert _{F _{SO} (M)} to F _{SO} (M)\to M of its natural lift \hat{X} onto FM\to M splits into a { S O}(n) -invariant vector field X _{K} on F _{SO} (M) , called the Kosmann lift of X , and a transverse vector field X _{G} .

See also

• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Spin bundle
• Connection (mathematics)
• G-structure
• Spin manifold
• Spin structure

Notes and References

1. Book: Fatibene, L. . Ferraris . M. . Francaviglia . M. . Godina . M. . 1996 . A geometric definition of Lie derivative for Spinor Fields . Proceedings of the 6th International Conference on Differential Geometry and Applications, August 28th–September 1st 1995 (Brno, Czech Republic) . Janyska . J. . Kolář . I. . Slovák . J. . Masaryk University . Brno . 549–558 . 80-210-1369-9 . gr-qc/9608003v1 . 1996gr.qc.....8003F .
2. Godina . M. . Matteucci . P. . 2003 . Reductive G-structures and Lie derivatives . . 47 . 66–86 . 10.1016/S0393-0440(02)00174-2 . math/0201235 . 2003JGP....47...66G .
3. (Example 5.2) pp. 55-56