Filed of Cones on (k, r)-Covelocities Vector Bundle on a Manifold

Constantin Pătrăşcoiu

Abstract

The study of the some important fields of geometrical objects (Riemannian or Lorenzian metrics) leads to some fields of cones defined on the tangent bundles.

In this Note, field of cones on (k,r)-covelocities vector bundle on a manifold and special case of r-th order tangent vector bundle are studied.

Mathematics Subject Classification: 53C65, 58A25 Key Words: currents, field of cones

Dennis Sullivan ([8],1976) associated to a field of convex and compact cones (cone of structure) in the bundles of tangent p-vectors on a manifold M a field of cones (cone structure) of currents (linear functional on C^{∞}-forms) generated by the Dirac currents.

The currents are ”directed” by an a-priori given field of cones in the space of tangent p-vectors and a positivity condition leads to a compact convex cone of currents with a compact convex subcone of cycles (closed currents).

Dan I. Papuc ([5],1992) introduced the notion of field of cones in a regular vector bundle and studied the geometry of this kind of general structure with direct application to the particular case of the tangent bundle to a maniflod (see also [2]-1988, [3]-1988, [4]-1990). Definition 1. (Dennis Sullivan [8], 1976) A compact convex cone C in a (locally convex topological) vector space V over R is a convex cone which for some (continuous) linear functional L : V → R satisfies L(x) > 0, ∀x ∈ C, x ≠ 0 and L^{−1} ∩ C is compact.

The set L^{−1} ∩ C is called a base for the cone often identified with the set of rays in the cone, and the kernel of L is called a strictly supporting hyperplane of the cone C. Definition 2. A cone structure (field of cones) on a closed subset N of a manifold M is a map

C : x ∈ N → C_{x},

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 97-102 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

where C_{x} is a compact convex cone in the vector space A_{p}(x) of tangent p-vectors on M, x ∈ M.

Continuity of cones is defined by the Hausdorf metric:

d(K, K′) = max(sup_{x∈K} ρ (x, K′), sup_{x′∈K′} ρ (x′, K′))

K = L^{−1}(1) ∩ C, K′ = L^{′−1} ∩ C′,

where ρ is a convenient metric on rays in some local trivialization of Λ_{p}. Definition 3. A differential p-form ω (of class C^{∞}) on M is transversal to the cone structure C if

∀ ν ≠ 0, ν ∈ C_{x}, (x ∈ N) ⇒ ω (ν) > 0.

Definition 4. A current (linear functional on C^{∞}-forms) is Dirac current if is one determined by the evaluation of p-forms on a single p-vector at one point. Definition 5. The cone of structure currents C^{∗} associated to the cone structure C is the closed convex cone of currents generated by the Dirac currents associted to elements of C_{x}, x ∈ N.

If N is compact the cone of structure currents C^{∗} associated to the cone structure C on N is a compact convex cone. Definition 6. (Dan I. Papuc [5], 1992) A field of cones on a regular vector bundle ξ = (E, p, M) (E,M paracompact connected manifolds without boundary) is a map

K : x ∈ M → K_{x} ⊂ E_{x} ⊂ E (E_{x} = p^{−1}(x))

such that the following two axioms to be satisfied:

1) ∀x ∈ M, K_{x} is a convex pointed closed cone having interior points in the space E_{x}).

2) The set ⋃_{x∈M} IntK_{x} and ⋃_{x∈M} Int(E_{x} − K_{x}) are open subsets of E. The pair (E_{x}, K_{x}) is a Krein space, the norm of E_{x} is arbitrarily chosen, and

all considerations concerning Krein spaces are available for (E_{x}, K_{x}).

An example of structure (ξ, K) is the tangent vector bundle of a Lorentzian time-oriented manifold (K_{x} is the quadratic cone of non space -like time- oriented tangent vectors).

The geometry of structure (ξ, K) was studied in [4],[5],[6].

In [8] to a regular vector bundle endowed with a field of cones there are associated the r-th jet prolongation and the initial field of cones are lifted in a natural way. Remark 1. ([6], 1992, p.44) A regular vector bundle ξ = (E, π, M) has a field of cones if and only if there is a continuous global section of non-zero vectors

s : M → E, π ∘ s = 1_{M}, ∀x ∈ M, s(x) ≠ 0.

In the following we shall use for fields of cones the Definition 6 (with the supplementary condition as the section from previous remark to be of class C^{∞} if it is necessary) because such fields of cones also are fields of cones introduced by the Definition 1.

Field of Cones (k, r)-Covelocities Vector Bundle on a Manifold 99

Let M be a manifold. The space T_{k}^{r^{∗}}M = J^{r}(M, R^{k})_{0} (the set of all r-jets of M into R^{k} with target 0) is called the space of (k, r)-covelocities on M.

Since R^{k} is a vector space, the fibber bundle (T_{k}^{r^{∗}}M, π_{k}^{r^{∗}}, M) is a vector bundle which verifies

j_{x}^{r}f + j_{x}^{r}g =^{def} j_{x}^{r}(f + g)

j_{x}^{r}(λf) =^{def} λ(j_{x}^{r}f), ∀j_{x}^{r}f, j_{x}^{r}g ∈ E_{x} = (π_{k}^{r^{∗}})^{−1}(x), λ ∈ R, x ∈ M.

It is called the vector bundle of (k,r)- covelocities on M. Every local diffeomorfism ϕ : M → N is extended to a vector bundle morphism T_{k}^{r^{∗}}φ : j_{x}^{r} ∈ T_{k}^{r^{∗}}M → j_{φ(x)}^{r}(f ∘ φ^{−1}) ∈ T_{k}^{r^{∗}}N, where φ^{−1} is constructed locally.

In this sense T_{k}^{r^{∗}} : Man(m) → F V is a functor. (Man(m) is the category of m- dimensional manifolds whose morphisms- locall diffeomorfisms and F V is the category of vector bundle).

For k = r = 1 we obtain the construction of the cotangent bundle as a functor T_{1}^{1^{∗}} = T^{∗} on Man(m). Proposition 1. For every field of cones K on tangent bundle (T M, π, M) of the manifold M we can define a ”canonical” field of cones K_{k}^{1∗} on the vector bundle (T_{k}^{1∗}M, π_{k}^{1∗}, M). Proof. Every field of cones K on the tangent bundle (T M, π, M) defines a field of cones K^{∗} on cotangentfibre bundle (T^{∗}M, π^{∗}, M),

K^{∗} : x ∈ M → K^{∗}(x) ⊂ (T^{∗}M)_{x},
K^{∗}(x) ^{def} = {ω_{x}/ ω_{x} ∈ (T^{∗}M)_{x}, ω_{x}(X_{x}) ≥ 0, ∀X_{x} ∈ K(x)}.

By virtue of Remark 1 there is a global section of nonzero covectors

ω : x ∈ M → ω_{x} ∈ T^{∗}M,
ω_{x} = j_{x}^{1}f ∈ T^{∗}M = J^{1}(M, R)_{0}.

Let by f_{k} : x ∈ M → (f(x), f(x), ..., f(x)) ∈ R^{k}. Obviously, j_{x}^{1}f_{k} depends on the germ of f at x only.

Then ω^{k} : x ∈ M → ω_{x}^{k} = j_{x}^{1}f_{k} ∈ T_{k}^{1∗}M is a global section nonzero (1, k)

- covelocities and according to Remark 1 there is a field of cones K_{k}^{1∗} on vector bundle (T_{k}^{1∗}M, π_{k}^{1∗}, M). Proposition 2. to every field of cones K_{k}^{r∗} on the vector bundle (T_{k}^{r∗}M, π_{k}^{r∗}, M) we can associate a field of cones K_{k}^{p∗} (p ≤ r) on the vector bundle (T_{k}^{p∗}M, π_{k}^{p∗}, M). Proof. Indeed K_{k}^{p∗} : x ∈ M → K_{k}^{p∗} ^{def} = {j_{x}^{p}f/j_{x}^{r}f ∈ K_{k}^{r∗}(x)} is a field of cones on the vector bundle (T _{k}^{p∗}M, π_{k}^{p∗}, M). Definition 7. The field of cones K_{k}^{r∗} and C_{k}^{r∗} on the vector bundle (T_{k}^{r∗}M, π_{k}^{r∗}, M) are p-equivalent 0 < p ≤ r; p ∈ N) if the fields of cones induced by them on the vector bundle (T_{k}^{p∗}M, π_{k}^{p∗}, M) coincide.

Evidently if two fields of cones are p- equivalent they are s-equivqlent ,s < p ≤ r.

The greatest natural number p from the Definition 7 is called order of equvalence for the field of cones K_{k}^{r∗} and C_{k}^{r∗}.

The relation introduced by the Definition 7 determines a partition for fields of cones on the vector bundles (T_{k}^{p∗}M, π_{k}^{p∗}, M) in class of orders of equivalence.

We remark that for k = 1 the fibre of vector bundle (T_{1}^{r∗}M, π_{1}^{r∗}, M) = (T^{r∗}M, π^{r∗}, M) is an algebra, and the multiplication in every space T_{x}^{r∗}M is given by

j_{x}^{r}f · j2_{x}g ^{def} = j_{x}^{r}(f ∘ g).

The projection π_{r,r−1} : T^{r∗}M → T^{r−1∗}M is a linear morphism of vector bundles.

Let f_{1}, f_{2}, ...f_{r} be differentiable functions on M with values 0 at x ∈ M.

The r-jet at x of their product j_{x}^{r}(f_{1}f_{2}...f_{r}) depends only on J_{x}^{1}f_{1}, J_{x}^{1}f_{2}, ... J_{x}^{1}f_{r} and lies in ker(π_{r,r−1}).

We can extended uniquely the map

(f_{1}, f_{2}, ...f_{r}) ∈ T^{∗}M × T^{∗}M × ... × T^{∗}M → j_{x}^{r}(f_{1}f_{2}...f_{r}) ∈ T^{r∗}M

into a linear isomorphism of S^{r}T^{∗}M (the r-th symmetric tensor power of T^{∗}M) in ker(π_{r,r−1}).

So, we have the following exact sequence of vector bundles over M :

(1)  0 → S^{r}T^{∗}M → T^{r∗}M − π_{r,r−1} → T^{r−1∗}M → 0

Remark 2. To every field of cones K on tangent bundle (T M, π, M) we can define a field of cones on S^{r}T^{∗}M :

S^{r∗}K : x ∈ M → S^{r}K^{∗}(x) ∈ S^{r}T^{∗}M
S^{r∗}K(x) ^{def} = {ρ_{x}/ρ_{x} ∈ S^{r}T^{∗}M, ρ_{x}(X_{1x}, X_{2x}, ..., X_{rx}) ≥ 0,
∀X_{1x}, X_{2x}, ..., X_{rx} ∈ K(x)}.

Because the sequence (1) of vector bundle over M is exact we can associated to field of cones S^{r∗}K a field of cones on ker(ı_{r,r−1} ⊂ T^{r∗}M whose projection on (T^{r−1∗}M, π^{r−1∗}, M) is zero vector.

Let by T_{k}^{r▷}M = (T_{k}^{r∗}M)^{∗},the dual vector bundle of the (k, r)−covelocities bundle on M.

For every map φ : M → N we can define the linear map

j_{x}^{r}f ∈ (T_{k}^{r∗})_{f(x)} → j_{x}^{r}(f ∘ φ) ∈ (T_{k}^{r∗}M)_{x}.

Its dual map

(T_{k}^{r▷}φ)_{x} : (T_{k}^{r▷}M)_{x} → (T_{k}^{r▷}N)_{f(x)}

determines a functor T_{k}^{r▷} on Man (the category of manifold) with values in the category of vector bundles F V.

In the most important case k = l we shall write T_{k}^{r▷} = T^{r▷}. The element of T^{r▷}M are called r-th order tangent vectors on M.

We remark that for r = l we have TM = (T^{∗}M)^{∗}.

Dualizing the exact sequence (1), we obtain

(2)  0 → T^{r−1▷}M → T^{r▷}M → S^{r}T M → 0.

This shows that there is a natural injection of the (r − 1)-st order tangent vectors into the r-th order one.

Proposition 3. To every field of cones K on the tangent bundle (T Mπ, M) we can define a canonical field of cones K^{r} on the vector bundle

(T^{r▷}M, π^{r}, M).

Because of natural injection of the (r − 1)-st order tangent vectors into the r-th order one, for a section in tangent bundle T M we can associate to a section of non-zero vectors in vector bundle (T^{p−1▷}M, π^{p−1}, M) a section of non-zero vectors in the vector bundle (T^{p▷}M, π^{p}, M) (p = 2...r).

So, there is a section of non-zero vectors in (T^{r▷}M, π^{r}, M) and according with Remark 1 the above proposition is true.

For every map φ : M → N the following diagram commutes

(3)  \begin{matrix}0 ⟶ T^{r−1▷}M ⟶ T^{r▷}M ⟶ S^{r}T M ⟶ 0 \\ ↓ T^{r−1▷}φ  ↓ T^{r▷}φ  ↓ S^{r}T φ \\ 0 ⟶ T^{r−1▷}N ⟶ T^{r▷}N ⟶ S^{r}T N ⟶ 0 \end{matrix}

Remark 3. The functors T^{r−1▷} and T^{r▷} from Man (M) to F V are natural functors.

The comutativity of diagram (3) and the fact that the sequences are exact make possible the construct a natural canonical operators Θ : T^{r−1▷} ≫⇒ T^{r▷} from the natural functor T^{r−1▷} to the natural functor T^{r▷}.

If on the tangent bundle there is a field of cones and we consider on the vector bundle of r-th order tangent vectors the field of associated cones, the canonical operators are positive.

References

[1] D.I. Papuc, Partial orderings on differentiable manifolds; An. Univ. Timişoara, Seria Matematica, vol. XXVI, f.1 (1988), 55-73.

[2] D.I. Papuc, Partial orderings on differentiable manifolds; An. Univ. Timişoara, Seria St. Matematice, vol. XXVI, f.1 (1988), 55-73.

[3] D.I. Papuc, On the geometry of a differentiable manidold with a partial odering (or with a field of tangent cones); An. Univ. Timişoara, Seria matematica, vol. XXVI, f.2 (1988), 39-48.

[4] D.I. Papuc, Ordered tangent structures associated to a differentiable manifold with a regular field of tangent cones; Bull. Math. De la Soc. Sci. Math. De la Roumanie, t.34, (82), nr.2 (1990), 173-179.

[5] D.I. Papuc, Field of cones and positive operators on a vector bundle; An. Univ. Timişoara, Seria Matematica, vol. XXX, f.1 (1992), 39-58.

[6] D.I. Papuc, Field of cones on a tensor bundle and eratum, (to appear).

[7] C.Pătrăşcoiu, Field of cones on the r-th jet prolongation of a regular vector bundle, (to appear).

[8] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds; Inventiones Math., 36 (1776), 225-255.

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