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HARMONIC VECTOR FIELDS
VARIATIONAL PRINCIPLES AND DIFFERENTIAL GEOMETRY
By SORIN DRAGOMIR, DOMENICO PERRONE Elsevier
Copyright © 2012 Elsevier Inc.
All rights reserved.
ISBN: 978-0-12-416032-3
CHAPTER 1
Geometry of the Tangent Bundle
Contents
1.1. The Tangent Bundle 2
1.2. Connections and Horizontal Vector Fields 4
1.3. The Dombrowski Map and the Sasaki Metric 6
1.3.1. Preliminaries on Local Calculations 8
1.3.2. Isotropic Almost Complex Structures 11
1.3.3. Invariant Isotropic Complex Structures 20
1.4. The Tangent Sphere Bundle 26
1.5. The Tangent Sphere Bundle over a Torus 29
The scope of this chapter is to briefly review the basic facts in the geometry of the tangent bundle T(M) over a Riemannian manifold (M, g), such as nonlinear connections, the Dombrowski map and the Sasaki metric Gs. Remarkably T(M) also carries a natural almost complex structure J (arising from g) compatible to Gs and such that (T(M), J, Gs) is an almost Kähler manifold. The almost complex structure J (discovered by P. Dombrowski,) is rarely integrable (in fact only when the base Riemannian manifold is locally Euclidean) yet J appears to be but one of the many isotropic almost complex structures Jδ,σ built by R.M. Aguilar,. On the other hand the existence of an integrable isotropic almost complex structure only requires that (M, g) has constant sectional curvature and, if this is the case, the family Jδ,σ contains a large subfamily of complex structures (among which the invariant ones may be completely determined, cf. Theorem 1.20). When an almost complex structure Jδ,σ is non integrable the geometry of (T(M), Jδ,σ) is related to the properties of the twisted Dolbeau complex (a description of which is given in Appendix A of this book). Further information on the geometry of T(M) (over a semi-Riemannian manifold M) is furnished in Chapter 7. Chapter 1 also contains the calculation (due to G. Wiegmink,) of the Bruschlinsky group of a torus T2 endowed with an arbitrary Riemannian metric leading to the classification up to homotopy of the unit tangent vector fields on T2. For the classical aspects of the geometry of the tangent bundle over a Riemannian manifold (the Sasaki metric, the almost contact metric structure of the unit tangent bundle, etc.) the reader may also consult the books by K. Yano & S. Ishihara, and D.E. Blair.
1.1. THE TANGENT BUNDLE
Let M be a real n-dimensional C∞ manifold and π : T(M) -> M its tangent bundle. If (U, [??]1, ..., [??]n) is a local coordinate system on M then let (π-1(U), xi, yi) be the naturally induced local coordinates on T(M) i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence T(M) is a real 2n-dimensional C∞ manifold. We set
[partial derivative]i = [partial derivative]/[partial derivative]xi, [??]i = [partial derivative]/[partial derivative]yi, 1 ≤ i ≤ n,
for the sake of simplicity. For didactic reasons, and only through this section, we distinguish notationally between the local coordinates [??]i (defined on U) and xi (defined on π-1(U)). Let
σ0 : M -> T(M), σ0(x) = 0x [member of] Tx(M), x [member of] M,
be the zero section. Then σ0 : M -> T(M) is an embedding of M in the (total space of its) tangent bundle. For each tangent vector v [member of] T(M) the subspace Vv [equivalent to] Ker(dvπ) [subset] Tv(T(M)) is the vertical space at v. A tangent vector X [member of] Vv is a vertical vector. The assignment
V : v [member of] T(M) [??] Vv [subset] Tv([T(M))
is a C∞ distribution of rank n on T(M) and {[??]i : 1 ≤ i ≤ n} is a local frame of V defined on the open subset π-1(U).
Definition 1.1 V is called the vertical distribution on T(M). A vector field X [member of] V (i.e., Xv [member of] Vv for any v [member of] T(M)) is a vertical vvvvector field on T(M).
The vertical distribution is involutive (as it may be easily seen by using the local frame {[??]i : 1 ≤ i ≤ n}). Therefore, by the classical Frobenius theorem, V is completely integrable and its maximal integral manifold passing through v [member of] T(M) is the tangent space Tx(M) where x = π(v) [member of] M.
Let π-1 T(M) -> T(M) be the pullback by π of the tangent bundle T(M) -> M. Its total space π-1TM is a submanifold of the product manifold T(M) × T(M). The fibre over v [member of] T(M) is
(π-1 T(M))v = {v} × Tπ(v)(M).
Alternatively π-1 T(M) is the largest subset of T(M) × T(M) such that the diagram
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is commutative. Here [??] and p are the (restrictions to π-1TM of the) first and second canonical projections of the product manifold T(M) × T(M).
Definition 1.2 Let X : M -> T(M) be a tangent vector field on M. The cross-section [??] : T(M) -> π-1 T(M) defined by [??](v) = (v,Xπ(v)), for any v [member of] T(M), is called the natural lift of X.
Let Xi : π-1(U) -> π-1 T(M) be the natural lift of the (local) tangent vector field [partial derivative]/[partial derivative][??]i : U -> T(M). Then {Xi : 1 ≤ i ≤ n} is a local frame of the pullback bundle π-1 T(M) -> T(M) defined on the open set π-1(U).
By a customary language abuse, one may identify the tangent bundle T(M) and the vertical bundle. Of course T(M) -> M and V -> T(M) both have rank n yet different base manifolds and the precise statement is that there is a natural vector bundle isomorphism π-1 T(M) ≈ V. Indeed for any v [member of] T(M) and any X [member of] Tπ(v)(M) let C : (-ε, ε) -> T(M) be the curve given by
C(t) = v + tX, |t| < ε (ε > 0).
For each v [member of] T(M) let γv: (π-1 T(M))v -> Tv(T(M)) be the map given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let (U, [??]i) be a local coordinate system on M such that x = π(v) [member of] U. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In particular, γv is an R-linear isomorphism (π-1 T(M))v ≈ Vv.
Definition 1.3 The vector bundle isomorphism γ : π-1 T(M) -> V = Ker(dπ) is called the vertical lift.
For each section s : T(M) -> π-1 T(M) the vertical vector field γs is the vertical lift of s. Note that in general s may fail to be the natural lift of a vector field on M. The vertical lift XV : T(M) -> T(T(M)) of a tangent vector field X : M -> T(M) is the vertical lift XV [equivalent to] γ[??] of the natural lift [??] of X.
1.2. CONNECTIONS AND HORIZONTAL VECTOR FIELDS
Let [nabla] be a linear connection on M. For any local coordinate system (U, xi) on M let Γijk : U -> R be the local connection coefficients i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Any linear connection [nabla] on M induces a connection [??] in the vector bundle π-1 T(M) -> T(M). This is easiest to describe locally, as follows. Let (U, xi) be a local coordinate system on M and let (π-1(U), xi, yi) be the induced local coordinates on T(M). We set by definition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
It may be easily checked that the definition doesn't depend upon the choice of local coordinates on M and that (1.1) gives rise to a (globally defined) connection in the vector bundle π-1 T(M) -> T(M).
Definition 1.4 A C∞ distribution H on T(M) is called a nonlinear connection on T(M) if
Tv(T(M)) = Hv [direct sum] Vv, v [member of] T(M), (1.2)
where V [equivalent to] Ker(dπ) is the vertical distribution.
A nonlinear connection H on T(M) is also referred to as a horizontal distribution on T(M) while H -> T(M) is the corresponding horizontal bundle. By (1.2) any horizontal distribution on T(M) has rank n.
Let us consider the bundle morphism L : T(T(M)) -> π-1 T(M) given by
LvA = (v, (dvπ)A), A [member of] Tv(T(M)), v [member of] T(M).
Locally
L([partial derivative]i) = Xi, L ([??]i) = 0, 1 ≤ i ≤ n.
Moreover the following sequence of vector bundles and vector bundle morphisms
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is exact. In particular given a nonlinear connection H on T(M) the restriction of L to H gives a vector bundle isomorphism L : H -? π-1 T(M) whose inverse is denoted by β : π-1 T(M) -> H.
(Continues...)
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