Lobachevskii Journal of Mathematics Vol.3, 1999, 97–111

\begin{matrix}c \\ ◯ \end{matrix}M. Graev

M. Graev

INTERMEDIATE INVERSION FORMULAS IN INTEGRAL GEOMETRY

(submitted by B.N.Shapukov)

Abstract. We solve the following “intermediate” problem of integral geometry: to reconstruct integrals of a function over p-dimensional planes in ℝ^{n} starting from its integrals over k-planes, where p < k. Some generalizations are also presented.

Introduction

The object of this article is an integral transformation that we denote by I_{k}. To every function f on ℝ^{n} this transformation assigns its integrals over k-dimensional planes, i.e., the function I_{k}f on the manifold H_{k} of all kdimensional planes. One of the principal problems of integral geometry is the reconstruction of the function f starting from the function I_{k}f. In [GGS1] the solution of this problem is reduced to the construction of an operator κ_{k}. To any function ϕ on H_{k} this operator assigns the differential form κ_{k}ϕ of degree k on H_{k}. It was proved that if ϕ = I_{k}f, then for every x ∈ ℝ^{n} the differential form κ_{k}ϕ is closed on the submanifold H^{x} ⊂ H_{k} of k-planes containing the point x, and

∫_{γ^{x}} κ_{k}ϕ = c(γ^{x}) f(x)  (1)

for every k-dimensional cycle γ^{x} ⊂ H^{x}, where c(γ^{x}) depends on the homology class of γ^{x} only. If k is even, then there exists a cycle γ^{x} such that c(γ^{x}) ≠ 0; thus ( 1) is the desired inversion formula. If k is odd, then c(γ^{x}) = 0 for every cycle γ^{x}.

Note that the operator κ_{k} is also important in other problems of integral geometry and its applications.

The aim of this article is to solve “intermediate” problems of integral geometry: it is required to reconstruct the integrals I_{p}f of a function f over p-dimensional planes starting from ϕ = I_{k}f, where p < k.

1991 Mathematical Subject Classification.44A05, 44A12, 44A90.

Key words and phrases.integral geometry, inversion formulas, intermediate problems, differential forms, intermediate differential forms.

We shall prove that for every p < k there exists an operator κ_{pk} from the space of functions on H_{k} into the space of differential (k − p)-forms on H_{k} with the following properties. If ϕ = I_{k}f, then for every p-plane h ⊂ ℝ^{n} the differential form κ_{pk}ϕ is closed on the submanifold H^{h} ⊂ H_{k} of k-planes containing h, and

∫_{γ^{h}} κ_{pk}ϕ = c(γ^{h}) I_{p}f(h)  (2)

for every (k − p)-cycle γ^{h} ⊂ H^{h}. If k − p is even, then there exists a cycle γ^{h} such that c(γ^{h}) ≠ 0.

In §1 we define the operators κ_{pk} in an invariant form. This construction has an inconvenience: both the functions I_{k}f and differential forms κ_{pk}ϕ are defined not on H_{k} but, generally speaking, on different vector bundles over H_{k}. This complicates the description of relationships between them. Therefore, in what follows (§2) all the definitions and constructions are given in local coordinates on H_{k}.

We shall prove that in addition to κ_{k}ϕ and κ_{pk}ϕ, there exists a large family of differential forms on H_{k} possessing properties similar to those of κ_{k}ϕ and κ_{pk}ϕ. These forms are constructed by means of another method in [GGS2] for the projective space ℙ^{n}, but their role in problems of integral geometry was not mentioned there. Note that the description of these forms in this article is much simpler than in [GGS2].

In §3 we generalize the results of §2 to integral transformations from functions on H_{p} to functions on H_{k}, where 0 ≤ p < k.

§1

In this section an invariant definition of intermediate inversion formulas is given.

1.1. Notation.

S(ℝ^{n}) is the Schwartz space of functions on ℝ^{n}.

E_{k} is the manifold of pairs (α, β), where α = (α_{1}, ⋯, α_{k}) is a k-frame in ℝ^{n}, β ∈ ℝ^{n}, k = 1, ⋯, n − 1.

H_{k} is the manifold of k-dimensional planes in ℝ^{n}.

H̃_{k} is the manifold of pairs (h, μ_{h}), where h ∈ H_{k} and μ_{h} is a measure on h invariant with respect to parallel translations. H̃_{k} is a one-dimensional fibering over H_{k}, because μ_{h} is defined uniquely up to coefficient.

The manifold E_{k} is endowed with the natural structure of a fibering over H̃_{k}:

E_{k} → H̃_{k}.

For every pair (α, β) ∈ E_{k} there is a corresponding element in H̃_{k}: the pair (h, μ_{h}), where the plane h ∈ H_{k} is given by the equation x = αt + β = α_{1}t^{1} + ⋯ + α_{k}t^{k} + β and dμ_{h} = dt^{1} ⋯ dt^{k}.

INTERMEDIATE INVERSION FORMULAS IN INTEGRAL GEOMETRY 99

1.2. The integral transformation I_{k}. Let us introduce an integral transformation of functions on R^{n}. To every function f ∈ S(ℝ^{n}) this transformation assigns the function I_{k}f on H̃_{k} defined by

(I_{k}f) (h, μ_{h}) = ∫_{h} f(x) dμ_{h}(x) = ∫_{ℝ^{k}} f(αt + β) d_{k}t,  (1.1)

where d_{k}t = dt^{1} ⋯ dt^{k}. It is evident that I_{k}f belongs to the Schwartz space S(H̃_{k}) of functions on H̃_{k}, which in addition satisfy the following condition:

(I_{k}f) (h, λμ_{h}) = λ (I_{k}f) (h, μ_{h})  (1.2)

for any λ > 0.

Usually we will consider the function I_{k}f in the lifting from H̃_{k} to E_{k} and write (I_{k}f) (α, β) instead of (I_{k}f) (h, μ_{h}). The condition that the function ϕ(α, β) = (I_{k}f) (α, β) can be dropped from E_{k} to H̃_{k} is equivalent to the following relations:

ϕ(αg, β + αt_{0}) = | det g|^{−1} ϕ(α, β)  (1.3)

for arbitrary g ∈ GL(k, ℝ) and t_{0} ∈ ℝ^{k}.

It is evident that for the function ϕ(α, β) = I_{k}f(α, β) the following equations hold:

\frac{∂^{2}ϕ}{∂β^{i}∂α_{s}^{j}} = \frac{∂^{2}ϕ}{∂β^{j}∂α_{s}^{i}} ,  \frac{∂^{2}ϕ}{∂α_{s}^{i}∂α_{t}^{j}} = \frac{∂^{2}ϕ}{∂α_{s}^{j}∂α_{t}^{i}}  (1.4)

for all s, t = 1, ⋯, k and i, j = 1, ⋯, n, where β^{i}, α_{s}^{j} are the coordinates of vectors β and α_{s}. Conversely [GGS1], if ϕ ∈ S(H̃_{k}) and ϕ satisfies the equations (1.4), then ϕ = I_{k}f, where f ∈ S(ℝ^{n}).

In [GGS1] a local inversion formula is derived which reconstructs the initial function f starting from the function ϕ = I_{k}f for every even k < n. Namely, an operator κ_{k} is introduced; to the function ϕ this operator assigns the differential k-form κ_{k}ϕ on H_{k} defined by

κ_{k}ϕ = ∑_{i_{1},··· ,i_{k}=1}^{n} \frac{∂^{k}ϕ}{∂β^{i_{1}} ⋯ ∂β^{i_{k}}} dα_{1}^{i_{1}} ∧ ⋯ ∧ dα_{k}^{i_{k}}.  (1.5)

For every x ∈ ℝ^{n} this form is closed on the submanifold H^{x} ⊂ H_{k} of the k-planes, containing the point x, and the following inversion formula is holds true:

∫_{γ^{x}} κ_{k}ϕ = c(γ^{x}) f(x)  (1.6)

for every k-dimensional cycle γ^{x} ⊂ H^{x}. Here c(γ^{x}) depends on the homology class of γ^{x} only, and there exists a cycle γ^{x} such that c(γ^{x}) ≠ 0.

In this section a representation of I_{p}f via I_{k}f is obtained for arbitrary p < k, where k − p is even.

1.3. Relationship between I_{k}f and I_{p}f for p < k.

Definition 1.1. Let us define an operator I_{kp} : S(E_{p}) → S(E_{k}) by the following formula:

\begin{aligned}(I_{kp}ϕ) & (α_{1}, ⋯, α_{k}; β) \\ & = ∫_{ℝ^{k−p}} ϕ(α_{k−p+1}, ⋯, α_{k}; α_{1}t^{1} + ⋯ + α_{k−p}t^{k−p} + β) dt^{1} ⋯ dt^{k−p}. \end{aligned}

(1.7)

Proposition 1.1. The following equation holds:

I_{kp} I_{p} f = I_{k} f  (1.8)

for every function f ∈ S(ℝ^{n}) and for arbitrary p, k, 0 ≤ p < k < n.

This equation follows from the definition of the operators I_{p}.

Let us describe the operator I_{kp} in an invariant form. Let (h, μ_{h}) ∈ H̃_{k}. Let us fix an arbitrary p-dimensional subspace l ⊂ ℝ^{n} parallel to h (i.e. l + x ⊂ h for every x ∈ h) and an arbitrary invariant measure μ_{l} on l. The measure μ_{l} induces the invariant measure on every p-plane h′ = l+x, x ∈ ℝ^{n} parallel to l. It is evident that the function on ℝ^{n}

ψ(x) = (I_{p}f) (l + x, μ_{l})

can be dropped on the factor space ℝ^{n}/l.

Let h̃ be an image of the k-plane h in the factor space ℝ^{n}/l ( i.e. h̃ is a (k − p)-plane), and let μh̃ be a measure on h̃ which is induced by a measure μ_{h} on h and a measure μ_{l} on l. Then we have: if

ϕ(h′, μ_{h′}) = (I_{p}f) (h′, μ_{h′}), (h′, μ_{h′}) ∈ H̃_{p},

then

(I_{kp}ϕ) (h, μ_{h}) = ∫h̃ ϕ(l + x, μ_{l}) dμh̃(x).  (1.9)

Remark. The equalities (1.7), (1.9) make sense only for the functions of H̃_{p} which belong to the image of I_{p}. If ϕ does not belong to the image of I_{p}, then the right-hand sides of (1.7) and (1.9) cannot be dropped to H̃_{p}.

1.4. Reconstruction of the function I_{p}f from the function I_{k}f for p < k.

Let H_{pk} → H̃_{p} be the fibering whose fiber over every (h, μ_{h}) ∈ H̃_{p} is the submanifold of all h′ ∈ H_{k} containing the p-plane h.

Let us define the following projection:

E_{k} → H_{pk};  (1.10)

the element (h, μ_{h}; h′) ∈ H_{pk} corresponds to every (α_{1}, ⋯, α_{k}; β) ∈ E_{k}, where h ∈ H_{p} is defined by the equation x = α_{k−p+1}t^{k−p+1} + ⋯ + α_{k}t^{k} + β, dμ_{h} = dt^{k−p+1} ⋯ dt^{k}, and h′ ∈ H_{k} is defined by the equation x = α_{1}t^{1} + ⋯ + α_{k}t^{k} + β.

Definition 1.2. To every function ϕ ∈ S(H_{k}) we assign the following differential (k − p)-forms κ_{pk}ϕ on E_{k}, p = 0, 1. · · · , k − 1:

κ_{pk}ϕ = ∑_{i_{1},··· ,i_{k−p}=1}^{n} \frac{∂^{k−p}ϕ}{∂β^{i_{1}} ⋯ ∂β^{i_{k−p}}} dα_{1}^{i_{1}} ∧ ⋯ ∧ α_{k−p}^{i_{k−p}}.  (1.11)

For p = 0 the form κ_{pk}ϕ coincides with the form κ_{k}ϕ from (1.5).

Proposition 1.2. The differential form κ_{pk}ϕ is dropped from E_{k} to H_{pk} under the mapping (1.10).

This proposition follows from the relations (1.3) for ϕ.

Proposition 1.3. If ϕ = I_{k}f, then the differential form κ_{pk}ϕ is closed on every fiber H_{pk}^{h} of the fibering H_{pk} → H̃_{p}.

Proposition 1.4. If ϕ = I_{k}f, then the following equation holds:

∫_{γ^{h,μ_{h}}} κ_{pk}ϕ = c(γ^{h,μ_{h}}) (I_{p}f) (h, μ_{h}),  (1.12)

for every (h, μ_{h}) ∈ H̃_{p} and every (k − p)-dimensional cycle γ^{h,μ_{h}} in the fiber H_{pk}^{h} over (h, μ_{h}). Here c(γ^{h,μ_{h}}) depends only on the homology class of γ^{h,μ_{h}}, and if k − p is even, then there exists a cycle γ^{h,μ_{h}} such that c(γ^{h,μ_{h}}) ≠ 0.

This equation follows from the inversion formula (1.6), where it is enough to replace k by k − p, and from proposition 1.1.

Remark. If k is even, then the function f ∈ S(ℝ^{n}) can be reconstructed from ϕ = I_{k}f with the help of the local inversion formula (1.6). For odd k, the reconstruction of f from I_{k}f can be reduced to two problems: to reconstruct I_{1}f from the function I_{k}f and to reconstruct f from the function I_{1}f.

By virtue of proposition (1.4), the first problem is solved by using local inversion formula (1.12). Thus all nonlocal inversion formulas for integral transformations I_{k}, where k is odd, are reduced to a single one: the reconstruction of functions f ∈ S(ℝ^{n}) from their integrals over straight lines.

§2

In this section the operators I_{k} and the intermediate inversion formulas are constructed in local coordinate systems on manifolds of k-planes in ℝ^{n}.

2.1.Integral transform I_{k}. Let M_{l,k+1} be the space of real l × (k + 1)- matrices

α = ∥α_{i}^{j}∥_{\begin{matrix}i=1,··· ,l \\ j=1,··· ,k+1 \end{matrix}}.

To every matrix α ∈ M_{l,k+1} we assign the k-plane in ℝ^{n} (n = k + l) which is given by the following equations:

x_{i} = ∑_{j=1}^{k} α_{i}^{j}x_{l+j} + α_{i}^{k+1},  i = 1, ⋯, l.  (2.1)

We choose the elements of the matrix α to be local coordinates in the manifold H_{k} of the k-planes in ℝ^{n}.

Let us introduce the integral transformation I_{k}. This transformation assigns to functions f ∈ S(ℝ^{n}) their integrals over k-dimensional planes in ℝ^{n}:

I_{k}f(α) = ∫_{ℝ^{n}} f(x) ∏_{i=1}^{l} δ(x_{i} − ∑_{j=1}^{k} α_{i}^{j}x_{l+j} − α_{i}^{k+1}) d_{n}x,  (2.2)

where δ(·) is the delta-function on ℝ and d_{n}x = dx_{1} ⋯ dx_{n}.

Proposition 2.1. The function ϕ = I_{k}f satisfies the following equations:

\frac{∂^{2}ϕ}{∂α_{i}^{j}∂α_{i′}^{j′}} = \frac{∂^{2}ϕ}{∂α_{i}^{j′}∂α_{i′}^{j}}  (2.3)

for all i, i′ = 1, ⋯, l and j, j′ = 1, ⋯, k + 1.

Each of the functions ψ = I_{k}f of §1 and ϕ = I_{k}f of (2.2) can be easily expressed in terms of the other. Namely,

ψ(α_{1}, ⋯, α_{k}; β) = ϕ(α),

if α_{i} = (α_{1}^{i}, ⋯, α_{l}^{i}, δ_{1}^{i}, ⋯, δ_{k}^{i}), i = 1, ⋯, k and β = (α_{1}^{k+1}, ⋯, α_{l}^{k+1}, 0, ⋯, 0).

2.2. Relatively closed differential k-forms. The differential form κ_{k}ϕ of §1 has the following form in local coordinates α_{i}^{j}:

κ_{k}ϕ = ∑_{s_{1},··· ,s_{k}=1}^{l} \frac{∂^{2}ϕ}{∂α_{s_{1}}^{k+1}, ⋯, ∂α_{s_{k}}^{k+1}} dα_{s_{1}}^{1} ∧ ⋯ ∧ dα_{s_{k}}^{k}.  (2.4)

This form is defined on the manifold H_{k} of k-planes over ℝ^{n}. If ϕ = I_{k}f, then κ_{k}ϕ is closed on the submanifold H^{x} ⊂ H_{k} of the k-planes containing an arbitrary point x ∈ ℝ^{n}. We shall call this property of κ_{k}ϕ the relative closedness.

We have according to §1:

∫_{γ^{x}} κ_{k}ϕ = c(γ^{x}) f(x)  (2.5)

for every x ∈ ℝ^{n} and for arbitrary k-dimensional cycle γ^{x} ⊂ H^{x}.

We shall show that, in addition to the form κ_{k}ϕ, there exist other relatively closed k-differential forms κ_{i}^{j}ϕ for which an analog of the inversion formula (2.5) holds.

Let us define

κ_{i}^{j}ϕ = κ_{i_{1},··· ,i_{k}}^{j_{1},··· ,j_{k}}ϕ = ∑_{s_{1},··· ,s_{k}=1}^{l} \frac{∂^{k}ϕ}{∂α_{s_{1}}^{i_{1}} ⋯ ∂α_{s_{k}}^{i_{k}}} dα_{s_{1}}^{j_{1}} ∧ ⋯ ∧ dα_{s_{k}}^{j_{k}}.  (2.6)

for all j = (j_{1}, ⋯, j_{k}) and i = (i_{1}, ⋯, i_{k}), where i_{s}, j_{s} = 1, ⋯, k + 1. In particular κ_{k+1,··· ,k+1}^{1,··· ,k}ϕ = κ_{k}ϕ.

Proposition 2.2. If the function ϕ(α) on H_{k} satisfies the equations (2.3), then the differential form κ_{i_{1},··· ,i_{k}}^{j_{1},··· ,j_{k}}ϕ is symmetric relative to i_{1}, ⋯, i_{k} and antisymmetric relative to j_{1}, ⋯, j_{k}.

Therefore, it is possible to assume that

j_{1} < j_{2} < ⋯ < j_{k}, i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{k}.  (2.7)

Proposition 2.3. If the function ϕ(α) on H_{k} satisfies the equations (2.3), then the differential forms κ_{i}^{j}ϕ are closed on the submanifolds H^{x} ⊂ H_{k} of all k-planes containing an arbitrary point x ∈ ℝ^{n}. If ϕ = I_{k}f, then

∫_{γ^{x}} κ_{i}^{j}ϕ = c(γ^{x}) P_{i}^{j}(x) f(x)  (2.8)

for an arbitrary k-dimensional cycle γ^{x} ⊂ H^{x}. Here c(γ^{x}) depends on the cohomology class of γ^{x} only; P_{i}^{j}(x) = ∏_{s=1}^{k} x_{l+i_{s}} if j = (1, ⋯, k) and P_{i}^{j}(x) = (−1)^{k−r}x_{l+r} ∏_{s=1}^{k} x_{l+i_{s}} if j = (1, ⋯, r−1, r+1, ⋯, k+1) , where x_{n+1} = 1. If k is even, then there exists a cycle γ^{x} such that c(γ^{x}) ≠ 0.

This proposition is a special case of Theorems 2.1 and 2.2, see 2.4 and 2.5.

2.3. Intermediate differential forms. Here we shall construct the analogues of differential forms κ_{pk}ϕ from §1. Let us introduce operators

d_{i}^{j} : Ω^{s} → Ω^{s+1},  (2.9)

where Ω^{s} is the space of differential s-forms on H_{k}. By definition,

d_{i}^{j} = ∑_{s=1^{l}} \frac{∂}{∂α_{s}^{i}} dα_{s}^{j},  i, j = 1, ⋯, k + 1.  (2.10)

Set

κ_{i_{1},··· ,i_{p}}^{j_{1},··· ,j_{p}} = ⋀_{s=1}^{p} d_{i_{s}}^{j_{s}},  s = 1, 2, · · · .  (2.11)

It is evident that

d(κ_{i_{1},··· ,i_{p}}^{j_{1},··· ,j_{p}}) = ∑_{s=1}^{k+1} κ_{s,i_{1},··· ,i_{p}}^{s,j_{1},··· ,j_{p}}.  (2.12)

It is evident also that if the function ϕ(α) on H_{k} satisfies the equations (2.3), then the differential forms κ_{i_{1},··· ,i_{p}}^{j_{1},··· ,j_{p}}ϕ are equal to zero for p > k + 1 and are closed on H_{k} for p = k + 1. The case p = k was considered in 2.2. Here we consider the differential forms κ_{i_{1},··· ,i_{p}}^{j_{1},··· ,j_{p}}ϕ for p < k. We shall call them the intermediate differential forms.

Note that the (k-p)-forms κ_{k+1,··· ,k+1}^{1,··· ,k−p}ϕ coincide with the forms κ_{pk}ϕ of §1, which are given in local coordinates.

If the function ϕ on H_{k} satisfies the equations (2.3), then, as in the case p = k, the forms κ_{i_{1},··· ,i_{p}}^{j_{1},··· ,j_{p}}ϕ are symmetric relative to i_{1}, ⋯, i_{p} and antisymmetric relative to j_{1}, ⋯, j_{p}. Therefore it is possible to assume that

j_{1} < ⋯ < j_{p},  i_{1} ≤ ⋯ ≤ i_{p}.  (2.13)

2.4. Relative closedness of intermediate forms.

Theorem 2.1. If the function ϕ on H_{k} satisfies the equations (2.3), then for every h ∈ H_{p}, p < k the differential (k − p)-forms κ_{i_{1},··· ,i_{k−p}}^{j_{1},··· ,j_{k−p}}ϕ are closed on the submanifold H^{h} ⊂ H_{k} of all k-planes containing the plane h, and κ_{i_{1},··· ,i_{s}}^{j_{1},··· ,j_{s}}ϕ ≡ 0 on H^{h} for s > k − p.

First, we shall find the incidence condition for h_{k} ∈ H_{k} and h_{p} ∈ H_{p} for p < k.

Proposition 2.4. Suppose that a k-plane h_{k} and a p-plane h_{p} in ℝ^{n}, p < k are given by an l×(k+1) matrix α and a q×(p+1) matrix β respectively (l = n − k, q = n − p). Then their incidence condition is

β_{j}^{i} = ∑_{s=1}^{q−l} α_{j}^{s} β_{l+s}^{i} + α_{j}^{q−l+i}, i = 1, ⋯, p + 1, j = 1, ⋯, l.  (2.14)

Let us write this condition in matrix form. We shall represent the matrices α and β as

α = (α^{l,q−l}, α^{l,p+1}),  β = ( \begin{matrix}β^{l,p+1} \\ β^{q−l,p+1} \end{matrix} ) ,  (2.15)

where γ^{r,s} denotes an r × s matrix. Then the condition of incidence (2.14) is

β^{l,p+1} = α^{l,q−l} β^{q−l,p+1} + α^{l,p+1}.  (2.16)

Remark. The set of matrices β satisfying condition (2.14) is a (q − l)(p + 1)-dimensional plane in the coordinate space ℝ^{N} (N = q(p + 1)) of all q × (p + 1) matrices β.

We shall now prove the theorem. Let

ω^{i} = (dα_{1}^{i}, ⋯, dα_{l}^{i}), i = 1, ⋯, k + 1.

It follows from (2.14) that we have on H^{h}:

ω^{j} + ∑_{i=1}^{k−p} β_{l+i}^{j} ω^{i} = 0,  j = k − p + 1. · · · , k + 1.  (2.17)

Thus there exist at most k − p linearly independent vectors ω^{i} on H^{h}, and therefore κ_{i_{1},··· ,i_{s}}^{j_{1},··· ,j_{s}}ϕ ≡ 0 on H^{h} for s > k − p. It follows from this remark and from (2.12) that the forms κ_{i_{1},··· ,i_{k−p}}^{j_{1},··· ,j_{k−p}}ϕ are closed on H^{h}.

2.5. Intermediate inversion formulas. Let κ_{i}^{j} = κ_{i_{1},··· ,i_{k−p}}^{j_{1},··· ,j_{k−p}}, p < k, where i_{s}, j_{s} = 1, ⋯, k + 1 and j_{1}, ⋯, j_{k−p} are pairwise distinct, and let us denote:

f_{i}(x) = f_{i_{1},··· ,i_{k−p}}(x) = (∏_{s=1}^{k−p} x_{l+i_{s}}) f(x) (x_{n+1} = 1).

Theorem 2.2. If ϕ = I_{k}f, then for every p-plane h ⊂ ℝ^{n} with coordinates β, the following intermediate inversion formula holds:

∫_{γ^{h}} κ_{i}^{j}ϕ = c(γ^{h}) Δ^{j}(β) (I_{p}f_{i})(h),  (2.18)

where γ^{h} is an arbitrary (k − p)-cycle in the submanifold H^{h} ⊂ H_{k} of kplanes containing the p-plane h, Δ^{j}(β) is the homogenous polynom of β and c(γ^{h}) depends on the homology class of γ^{h} only. If k − p is even, then there exists a cycle γ^{h} such that c(γ^{h}) ≠ 0.

Proof. Set

∂_{i} = (\frac{∂}{∂α_{1}^{i}} , ⋯ \frac{∂}{∂α_{l}^{i}} ) , i = 1, ⋯, k + 1;

thus d_{i}^{j} = ⟨∂_{i}, ω^{j}⟩. It follows from the definition of I_{k} that

∂_{s}(I_{k}f) = ∂_{k+1}(I_{k}f_{s}), s = 1, ⋯, k,

where f_{s}(x) = x_{l+s} f(x). Consequently,

κ_{i}^{j}I_{k}f = κ_{k+1,··· ,k+1}^{j_{1},··· ,j_{k−p}}I_{k}f_{i_{1},··· ,i_{k−p}}.  (2.19)

Further, it follows from (2.17) that

κ_{k+1,··· ,k+1}^{j_{1},··· ,j_{k−p}}ϕ = Δ^{j_{1},··· ,j_{k−p}}(β) κ_{k+1,··· ,k+1}^{1,··· ,k−p}ϕ  (2.20)

on H^{h}, where Δ^{j} is a homogenous polynom in the coordinates β of h. According to §1,

∫_{γ^{h}} κ_{k+1,··· ,k+1}^{1,··· ,k−p}I_{k}f = c(γ^{h}) I_{p}f(h)

for every (k − p)-cycle γ^{h} ⊂ H^{h}, and if k − p is even, then there exists a cycle γ^{h} for which c(γ^{h}) ≠ 0. It follows from this and from (2.19), (2.20) that (2.18) holds.

2.6. Dual intermediate forms. Set

κ̂_{i_{1},··· ,i_{q}}^{j_{1},··· ,j_{q}} = ⋀_{s=1}^{q} d̂_{i_{s}}^{j_{s}},  i_{s}, j_{s} = 1, ·, l,  (2.21)

where

d̂_{i_{s}}^{j_{s}} = ∑_{r=1}^{k+1} \frac{∂}{∂α_{j_{s}}^{r}} dα_{i_{s}}^{r}.  (2.22)

We call the operators κ̂_{i_{1},··· ,i_{q}}^{j_{1},··· ,j_{q}} dual to the operators of p. 2.4.

We have by analogy with Theorem 2.1:

Theorem 2.3. If the function ϕ on H_{k} satisfies the equations (2.3), then for every p = k + 1, ⋯, n − 1, the differential (p − k)-forms κ̂_{i_{1},··· ,i_{p−k}}^{j_{1},··· ,j_{p−k}}ϕ are closed on the submanifolds of all k-planes belonging to an arbitrary p-plane h, and κ_{i_{1},··· ,i_{s}}^{j_{1},··· ,j_{s}}ϕ ≡ 0 on these submanifolds for s > p − k.

We shall now explain the role of the differential forms κ̂_{i_{1},··· ,i_{p−k}}^{j_{1},··· ,j_{p−k}}ϕ in the inversion formulas. We shall speak for convenience of “the matrix” instead of “the plane”, because the manifolds H_{k} of k-planes in ℝ^{n} are realized in local coordinates as spaces of l × (k + 1)-matrices. Between the matrices of H_{k} and H_{p} the incidence relation is defined which is given by formula (2.14).

Let us define the mappings σ = σ_{pk:} H_{p} → H_{q−1} and H_{k} → H_{l−1} (p + q = k + l = n) obtained by the permutation of matrix blocks in (2.15), alteration of the sign and transposition. It is evident that if the matrices α ∈ H_{k} and β ∈ H_{p} are incident, then the matrices σα ∈ H_{l−1} and σβ ∈ H_{q−1} are also incident.

The mapping σ : H_{k} → H_{l−1} induces a bijection between the functions on H_{k} and H_{l−1}; it associates the function σψ(α) = ψ(σα) on H_{k} to every function ψ on H_{l−1}. It is evident that if the function ψ on H_{l−i} satisfies the equations (2.3), then the function σψ on H_{k} also satisfies the equations (2.3).

Theorem 2.4. Let ϕ = σψ, where ψ is a function on H_{l−1}. If ψ = I_{l−1}f for a function f ∈ S(ℝ^{n}), then for every p-plane h ⊂ ℝ^{n}, p > k, the following equality holds:

∫_{γ_{h}} κ̂_{i_{1},··· ,i_{p−k}}^{j_{1},··· ,j_{p−k}} ϕ = c(γ^{h}) Δ^{j}(h) I_{q−1}f_{i}(σh), q = n − p.

Here γ^{h} is an arbitrary (p−k)-cycle in the manifold H^{h} ⊂ H_{k} of all k-planes belonging to the p-plane h; the functions Δ^{j} and f_{i} are defined as in Theorem 2.2. If p − k is even, then there exists a cycle γ^{h} such that c(γ^{h}) ≠ 0.

2.7. Example. Let us consider the integral transformation ϕ = I_{1}f in ℝ^{4}. By definition, we have

ϕ(α) = ∫_{−∞}^{+∞} f(α_{1}^{1}x_{4} + α_{1}^{2}, α_{2}^{1}x_{4} + α_{2}^{2}, α_{3}^{1}x_{4} + α_{3}^{2}, x_{4}) dx_{4},

where α is a 3 × 2 matrix.

The following differential forms are closed on the submanifolds of all lines in ℝ^{4} containing an arbitrary fixed point:

κ_{i}^{j}ϕ = ∑_{s=1}^{3} \frac{∂ϕ}{∂α_{s}^{i}} dα_{s}^{j}, i, j = 1, 2.

The following differential forms are closed on the submanifolds of all lines belonging to an arbitrary fixed 2-plane:

κ̂_{i}^{j}ϕ = ∑_{s=1}^{2} \frac{∂ϕ}{∂α_{j}^{s}} dα_{i}^{s}, i, j = 1, 2, 3.

The following differential forms are closed on the submanifolds of all lines belonging to an arbitrary fixed hyperplane:

κ̂_{i_{1},i_{2}}^{j_{1},j_{2}}ϕ = ∑_{s_{1},s_{2}=1}^{2} \frac{∂^{2}ϕ}{∂α_{j_{1}}^{s_{1}} ∂α_{j_{2}}^{s_{2}}} dα_{i_{1}}^{s_{1}} ∧dα_{i_{2}}^{s_{2}}, 1 ≤ i_{1} < i_{2} ≤ 3, 1 ≤ j_{1} ≤ j_{2} ≤ 3.

Let us define a function on the space of all 2 × 3 matrix by the equality

ψ(α) = ϕ(α′).

Then ψ = I_{2}F for some function F on ℝ^{4}. It can be easily verified that

F (x) = c ∫_{ℝ^{4}} f(y) δ′′(y_{1}x_{3} + y_{2}x_{4} + y_{3} − y_{4}x_{1} − x_{2}) d_{4}y.

§3

In this section, the results of §2 are generalized to the case of integral transformations from the functions on H_{p} to the functions on H_{k}, where 0 ≤ p < k.

3.1. Integral transformation I_{kp}. Suppose that the elements of H_{k} and H_{p}, where 0 ≤ p < k < n, are given by l × (k + 1) matrices α and q × (p + 1) matrices β respectively (l = n − k, q = n − p). We shall write these matrices in the form (2.15). Then the condition that the p-plane with the coordinates β is contained in the k-plane with the coordinates α is

β^{l,p+1} = α^{l,q−l} β^{q−l,p+1} + α^{l,p+1}.

Definition 3.1. We shall define a transformation I_{kp} from functions f(β) = f(β^{l,p+1}, β^{q−l,p+1}) on H_{p} to functions ϕ(α) = ϕ(α^{l,q−l}, α^{l,p+1}) on H_{k} by the following equation:

I_{kp}f(α^{l,q−l}, α^{l,p+1}) = ∫ f(α^{l,q−l}β^{q−l,p+1} + α^{l,p+1}, β^{q−l,p+1}) dβ^{q−l,p+1},

(3.1)

where dβ^{q−l,p+1} = ∏_{i,j} dβ_{i,j}^{q−l,p+1}, β_{i,j}^{q−l,p+1} are the matrix elements of β^{q−l,p+1}.

Thus, by definition, I_{kp}f(α) is the integral over the set of all p-planes contained in the k-plane with the coordinates α. In the case p = 0, the operator I_{kp} coincides with the operator I_{k} of §2.

Note that if p + k ≥ n, then the kernel of the transformation I_{kp} is nontrivial, because dim H_{k} < dim H_{p} in this case. Therefore we shall assume in what follows that p + k < n.

3.2. Differential form κϕ and inversion formula for I_{kp}. Let us denote the elements of the matrices α^{l,q−l} and α^{l,p+1} by α̃_{i}^{j} and α_{i}^{j′} respectively; i = 1, ⋯, l; j = 1, ⋯, q − l; j′ = 1, ⋯, p + 1.

Let us introduce the following operator on the space of functions f(α):

κ = ⋀_{i=1}^{p+1} ⋀_{j=1}^{k−p} d_{i}^{j},  (3.2)

where

d_{i}^{j} = ∑_{s=1}^{l} \frac{∂}{∂α_{s}^{i}} dα̃_{s}^{j}.  (3.3)

Thus, if ϕ is a function on H_{k}, then κϕ is the differential N_{kp}-form on H_{k}, where N_{kp} = (k − p)(p + 1). Note that in the case p = 0, the form κϕ coincides with the form κ_{k}ϕ of §2.

Theorem 3.1. If ϕ = I_{kp}f, then for every p-plane h, the differential form κϕ is closed on the submanifold H^{h} ⊂ H_{k} of all k-planes containing h, and

∫_{γ^{h}} κϕ = c(γ^{h}) f(h)  (3.4)

for every N_{kp}-dimensional cycle γ^{h} ⊂ H^{h}. If k − p is even, then there exists a cycle γ^{h} such that c(γ^{h}) ≠ 0.

This theorem was proved in [GGS3] (for the projective case). It is a special case of Theorem 3.2, see 3.3.

3.3. Intermediate inversion formulas. Let

j = ∥j_{iv}∥_{\begin{matrix}i=1,··· ,p+1 \\ v=1,··· ,k−r \end{matrix}}  (3.5)

be an arbitrary (p+1)×(k −r) matrix of natural numbers j_{iv} = 1, ⋯, k −p, where 0 ≤ p ≤ r < k < n. We shall assume that the numbers j_{i1}, ⋯, j_{i,k−r} are pairwise distinct for every i.

Definition 3.2. Let us define an operator κ_{j}, where j is a matrix of the form (3.5), by the following equation:

κ_{j} = ⋀_{i=1}^{p+1} ⋀_{v=1}^{k−r} d_{i}^{j_{iv}},  (3.6)

where the operators d_{i}^{j_{iv}} are given by the formula (3.3).

Thus, κ_{j}ϕ is a differential (k − r)(p + 1)-form on the manifold H_{k}. Note that if r = p and j_{iv} = v for every i = 1, ⋯, p + 1, then the form κ_{j}ϕ coincides with the form κϕ of 3.2, Theorem 3.2. If ϕ = I_{kp}f, then for every r-plane h, where p ≤ r < k, the differential form κ_{j}ϕ is closed on the submanifold H^{h} ⊂ H_{k} of all k-planes containing h, and

∫_{γ^{h}} κϕ = c(γ^{h}) I_{rp}f(h)  (3.7)

for every (k − r)(p + 1)-dimensional cycle γ^{h} ⊂ H^{h}. If k − r is even, then there exists a cycle γ^{h} such that c(γ^{h}) ≠ 0.

Proof. We shall view the space H_{p} of q × (p + 1) matrices β (q = n − p) as

ℝ_{1}^{q} × ⋯ × ℝ_{p+1}^{q}

where ℝ_{i}^{q} is the q-dimensional coordinate space with coordinates β_{1}^{i}, ⋯, β_{q}^{i}.

Let H_{s,r−p+1}^{i} (s = n − r) be the space of (r − p)-dimensional planes in ℝ_{i}^{q}, and let I_{r−p}^{i} (p ≤ r ≤ k) be the operator in the Schwartz space of functions on ℝ_{i}^{q} defined in §2, i = 1, ⋯, p + 1. To every function f on ℝ_{i}^{q} the operator I_{r−p}^{i} assigns its integrals over (r − p)-dimensional planes in ℝ_{i}^{q}, i.e. the function I_{r−p}^{i}f on the space H_{s,r−p+1}^{i}.

Set

H_{r} = H_{s,r−p+1}^{1} × ⋯ × H_{s,r−p+1}^{p+1}, p ≤ r ≤ k,

and

I_{r} = I_{r−p}^{1} × ⋯ × I_{r−p}^{p+1}, p ≤ r ≤ k.

By definition, I_{r} is the integral transformation in the space of functions on H_{p}. To every function f(β) this transformation assigns its integrals over all manifolds h^{1} × ⋯ × h^{p+1} ∈ H_{r}, i.e. the function ψ = I_{r}f on H_{r}.

Let us define operators κ̂_{i}^{j_{i1},··· ,j_{i,k−r}}, p ≤ r < k in the space of functions on H_{l,k−p+1}^{i} (l=n-k). If γ = ∥γ_{i}^{j}∥ are coordinates on H_{l,k−p+1}^{i}, then

κ̂_{i}^{j_{i1},··· ,j_{i,k−r}} = ⋀_{v=1}^{k−r}(∑_{u=1}^{l} \frac{∂}{∂γ_{u}^{k−r+1}} dγ_{u}^{j_{iv}}).

Set

κ̂_{j} = ⋀_{i=1}^{p+1} κ̂_{i}^{j_{i1},··· ,j_{i,k−r}}.

Thus, by definition, κ̂_{j}ψ is a differential (k − r)(p + 1)-form on H_{k} for every function ψ on H_{k}.

For every ĥ = h_{k−r}^{1} × ⋯ × h_{k−r}^{p+1} ∈ H_{r}, p ≤ r < k, we denote by Hĥ ⊂ H_{k} the subset of all h_{k−p}^{1} × ⋯ × h_{k−p}^{p+1} ∈ H_{k} such that h_{k−r}^{i} is contained in h_{k−p}^{i}, i = 1, ⋯, p + 1.

Proposition 3.1. If ψ = I_{k}f, then the differential form κ̂_{j}ψ on H_{k} is closed on the submanifolds Hĥ and

∫_{γĥ} κ̂_{j}ψ = c(γĥ) I_{r}f(ĥ)

for every (k − r)(p + 1)-dimensional cycle γĥ ⊂ Hĥ. If k − r is even, then there exists a cycle γĥ such that c(γĥ) ≠ 0.

This proposition follows from Theorems 2.1 and 2.2.

Let us define imbeddings

π_{r} : H_{r} ↪ H_{r}, p ≤ r ≤ k.

For every r-plane h ∈ H_{r} with coordinates γ = ∥γ_{i}^{j}∥, the imbedding π_{r} assigns an element π_{r}h = h_{k−r}^{1} × ⋯ × h_{k−r}^{p+1} ∈ H_{r}, where h_{r−p}^{i} is the (r − p)- plane in ℝ_{i}^{q} given by the following equations:

β_{j}^{i} = ∑_{u=1}^{r−p} γ_{j}^{u}β_{s+u}^{i} + γ_{j}^{r−p+i},  j = 1, ⋯, s  (s = n − r).

In the matrix form, if γ = (γ^{s,r−p}, γ^{s,p+1}), then

π_{r}γ = (γ^{s,r−p}, γ_{1}) × ⋯ × (γ^{s,r−p}, γ_{p+1}),

where γ_{i} denotes the i-th column of γ^{s,p+1}.

Proposition 3.2. We have for every h ∈ H_{r}, p ≤ r < k:

π_{r}H^{h} ⊂ H^{π_{r}h}.

This proposition follows from the incidence conditions for planes.

It is easily verified that if ϕ = I_{kp}f and ψ = I_{k}f, then ϕ = ψ|_{H_{k}} and κ_{j}ϕ = κ̂_{j}ψ|_{H_{k}}. Theorem 3.2 follows immediately from this and from Propositions 3.1, 3.2.

Remark. All the results of this article may be transferred to the case of the complex space ℂ^{n}. Note that for the complex space, the inversion formulas are valid without a restriction on the evenness.

Acknowledgements

This paper was supported by Erwin Schroedinger International Institute for Mathematical Physics and by the Russian Foundation for Basic Research (RFBR). The author would like to express his deep gratitude to I. Andreeva, G. Litvinov and M. Schwartz for their important help in the preparation of this article.

References

[GGG] Gelfand I., Gindikin S., Graev M., Integral geometry in affine and projective spaces., Itogi nauki i techniki, v. 16 (1980) VINITI, pp. 55–224 (Russian); Engl. transl.: J. Sov. Math., v. 18 (1980).

[GGV] Gelfand I., Graev M., Vilenkin N., Integral geometry and representation theory, Generalized Functions, vol. 5, Fizmatgiz, 1962 (Russian) ; Engl. transl.: Academic press, 1966.

[GGS1] Gelfand I., Graev M., Shapiro Z., Integral geometry on k-dimen-sional planes, Funct. Anal. Prilozh., v. 1 (1967), pp. 15–31 (Russian); Engl. transl.: Funct. Anal. Appl., v. 1 (1967), pp. 14–27.

[GGS2] Gelfand I., Graev M., Shapiro Z., Integral geometry in the projective space, Funct. Anal. Prilozh., v. 4 (1970), pp. 14–37 (Russian); Engl. transl.: Funct. Anal. Appl., v. 4 (1970), pp. 12–28.

[GGS3] Gelfand I., Graev M., Shapiro Z., A problem of integral geometry connected with a pair of Grassmann manifolds, Doklady Acad. Nauk, v. 193, No. 2 (1970), pp. 259–262 (Russian); Engl. transl.: Soviet Math. Doklady, v. 11, No. 4 (1970), pp. 892–896.

Research Institute for System Studies, Russian Academy of Sciences 23, Avtozavodskaya street, Moscow 109280, Russia

E-mail address: graev@niisi.msk.ru

Received July 27, 1999