Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 13–22

MEASURE OF LINEAR SPACES AND TOTAL CURVATURES

OF COMPACT ANTI–INVARIANT SUBMANIFOLDS IN I C^{n}.

Ximo Gual Arnau

Abstract. Let M be a totally real compact manifold, immersed in the Kaehler manifold IR^{2n}. We define the total curvatures of M and we get a local interpretation of them. Previously, we obtain an invariant measure for the different linear spaces in I C^{n}.

Keywords. Total curvatures, densities, subspaces in I C^{n}, moving frames.

MS Classification. 53C64.

0.- Introduction

The total absolute curvatures of compact differentiable manifolds immersed in the euclidean space IR^{n}, have been defined by L. A. Santaló as curvatures of global character in the manifold, [SA-1]; moreover, he expressed them as the integral of certain local curvatures defined in every point of the manifold, [SA-2].

In this work, we will consider totally real (anti-invariant) submanifolds in the complex space I C^{n}, or more specifically in IR^{2n} as Kaehler manifold [K-M], and following a similar process as in [SA-1, 2], we will define the total absolute curvatures of these submanifolds and we will present them as the integral over the submanifold of certain local curvatures. These definitions allow us to compare the new curvatures with the total curvatures for submanifolds in IR^{2n} as real euclidean space.

In order to obtain these goals we need some results of Integral Geometry for linear spaces in I C^{n}, which permit us to define a density (measure) of them, invariant under the group of complex motions. These measures will be obtained following the general method for densities in homogeneous spaces, [SA-3].

The linear spaces that we will consider are basically the isotropic, coisotropic and symplectic subspaces of (IR^{2n}, ω), where ω is the symplectic (Kaehler) form defined by the complex operator J, [L-R].

Typeset by AMS-TEX

The author was partially supported by the Fundaciò Caixa Castellò grant n. MI25043/92.

This paper is in final form and no version of it will be submitted for publication elsewhere .

1.- Review of hermitian geometry

From now on, I C^{n} will be identified with IR^{2n} and its canonical complex structure J, and we will use the canonical inner product in IR^{2n} as Hermitian inner product with respect to J.

In order to obtain the density for the different linear spaces in I C^{n}, we will use the method of moving frames, [SH].

1.1. Definition. A moving frame on an open set U ⊂ I C^{n} is a set (e_{1}(p), …, e_{n}(p), e_{1∗}(p), …, e_{n∗}(p)) of smooth IR^{2n}-valued functions such that for each p ∈ U, (e_{1}, …, e_{n}, e_{1∗}, …, e_{n∗}) is a unitary frame for IR^{2n} ≅ T_{p}IR^{2n}; where e_{i∗} = Je_{i}, 1 ≤ i ≤ n.

Let p : U ↪ IR^{2n} be the inclusion map, so that the vector valued 1-form dp is the identity at each point of U, i.e., dp (e_{i}) = e_{i}.

We define the following 1-forms:

(1.1)  ω_{i} = dp e_{i}, ω_{i∗} = dp e_{i∗}, 1 ≤ i ≤ n,

(1.2)  \begin{matrix}ω_{ij} = de_{i} e_{j}, ω_{i∗j∗} = de_{i∗} e_{j∗}, \\ ω_{i∗j} = de_{i∗} e_{j}, ω_{ij∗} = de_{i} e_{j∗}, \end{matrix} 1 ≤ i, j ≤ n.

1.2. Note. The 1-forms (ω_{i}, ω_{i∗}) are nothing but the dual basis of (e_{i}, e_{i∗}), since

ω_{i}(e_{j}) = dp e_{i}(e_{j}) = dp(e_{j})e_{i} = e_{j} e_{i} = δ_{ij}.

Taking this into account it is clear that ω_{ij} = ω_{i∗j∗} and ω_{i∗j} = −ω_{ij∗}, so we have the structure equations (1.3)

\begin{matrix}dω_{i} = ∑_{k=1}^{n} ω_{ik} ∧ ω_{k} + ∑_{k=1}^{n} ω_{ik∗} ∧ ω_{k∗}, & dω_{i∗} = ∑_{k=1}^{n} ω_{i∗k} ∧ ω_{k} + ∑_{k=1}^{n} ω_{ik} ∧ ω_{k∗}, \\ dω_{ij} = ∑_{k=1}^{n} ω_{ik} ∧ ω_{kj} + ∑_{k=1}^{n} ω_{ik∗} ∧ ω_{k∗j}, & dω_{ij∗} = ∑_{k=1}^{n} ω_{ik} ∧ ω_{kj∗} + ∑_{k=1}^{n} ω_{ik∗} ∧ ω_{k∗j∗}. \end{matrix}

1.3. Definition. A vector subspace π of IR^{2n} is said to be holomorphic relative to the complex structure J if π is invariant under J, i.e., Jπ = π.

A vector subspace π of IR^{2n} is said to be antiholomorphic or totally real if Jπ ⊂ π^{⊥}.

By L_{2r+t} we will denote a subspace of IR^{2n} that has a 2r-dimensional holomorphic section and a t-dimensional antiholomorphic section.

1.4. Note. If we consider the symplectic structure of IR^{2n} inherits from its complex structure we find that holomorphic subspaces correspond to symplectic subspaces, antiholomorphic subspaces to isotropic subspaces and L_{2r+t} with r + t = n to coisotropic sub-spaces; [L-R].

Total curvatures of real submanifolds in I C^{n} 15

1.5. Definition. A differentiable manifold M immersed in the euclidean space IR^{2n}, is called anti-invariant (totally real) submanifold of I C^{n} if

∀p ∈ M, J(T_{p}M) ⊂ (T_{p}M)^{⊥}.

An anti-invariant submanifold of IR^{2n} is isotropic with respect to the symplectic structure.

1.6. Note. The complex motions form a group M^{c} defined by the equations,

x′ = ax + b, a = A + iB ∈ U(n), b = b_{1} + ib_{2} ∈ I C^{n}.

2.- Densities for linear spaces in I C^{n}

In order to define the total curvature for anti-invariant submanifolds immersed in I C^{n}, we need, as in the real case, to define a measure of the different linear spaces introduced before, invariant by M^{c}.

These invariant densities will be obtained following the general method for homogeneous spaces, [SA-3].

2.1.- Density for holomorphic linear spaces.

We seek a density for holomorphic planes L_{2r} of real dimension 2r, invariant under the group M^{c}.

Let L_{2r}^{0} be a fixed holomorphic plane, spanned by the point p and the unit vectors (e_{1}, …, e_{r}, e_{1∗}, …, e_{r∗}); if we consider the subgroup of M^{c} formed by the complex motions that leave L_{2r}^{0} invariant, and that will be denoted by H_{2r}; evidently, we have a bijective correspondence between the set of holomorphic subspaces and the elements of the homogeneous space M^{c}/H_{2r}, therefore we find that the density for holomorphic planes L_{2r} will be the density on the homogeneous space M^{c}/H_{2r}.

Since L_{2r}^{0} must keep invariant under M^{c}, then

(2.1.1)  ω_{α} = ω_{α∗} = ω_{iα} = ω_{iα∗} = 0,  for 1 ≤ i ≤ r, r + 1 ≤ α ≤ n.

These forms are a completely integrable system and by general theory [SA-3], it is known that the invariant density for M^{c}/H_{2r} becomes

(2.1.2)  dL_{2r} = ⋀_{α}(ω_{α} ∧ ω_{α∗}) ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}),

where 1 ≤ i ≤ r, and r + 1 ≤ α ≤ n.

It is the unique form invariant under M^{c}, up to a constant factor.

Density for holomorphic planes about a fixed plane.

The density for linear holomorphic spaces which contain a fixed holomorphic space; that is, we have that L_{2q} is invariant and L_{2r} variable with L_{2q} ⊂ L_{2r}, will be equal to the invariant volume element of H_{2q}/H_{2r[2q]}; where H_{2q} is the group of complex motions that maintain a fixed L_{2q}^{0} invariant and H_{2r[2q]} is the subgroup of H_{2q} formed by the elements that keep a fixed L_{2r}^{0} invariant, with L_{2q}^{0} ⊂ L_{2r}^{0}. Following the procedure we have used before, this density is

(2.1.3)  dL_{2r[2q]} = ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}), with q + 1 ≤ i ≤ r; r + 1 ≤ α ≤ n.

2.1. Note. For q = 0 we get the density for holomorphic 2r-planes about a fixed point, which is the invariant volume element of the Grassmann manifold G_{r,n}^{c},

(2.1.4)  \frac{U(n)}{U(r) × U(n − r)}.

2.2. Note. The holomorphic planes L_{2r} around a totally real plane L_{q} with q < r, have the invariant density given by dL_{2r[2q]}; because J is an automorphism, and then, if we keep the vectors e_{i} fixed for 1 ≤ i ≤ q, the vectors e_{i∗} will also be fixed, for 1 ≤ i ≤ q.

2.2.- Density for totally real subspaces.

Using a similar procedure as in 2.1, the invariant volume element for totally real subspaces in I C^{n} will be the density on the homogeneous space M^{c}/H_{r}, where H_{r} is the subgroup of M^{c} whose elements are the complex motions that leave invariant a totally real plane L_{r} fixed, spanned by the vectors (e_{1}, …, e_{r}).

In order to keep L_{r} fixed, it is necessary that (2.2.1)

ω_{α} = ω_{α∗} = ω_{j∗} = ω_{iα} = ω_{iα∗} = ω_{ij∗} = 0, for 1 ≤ i ≤ j ≤ r, r + 1 ≤ α ≤ n.

And then, the density for anti-invariant linear spaces will be:

(2.2.2)  dL_{r} = ⋀_{α}(ω_{α} ∧ ω_{α∗}) ⋀_{j} ω_{j∗} ⋀_{i≤j} ω_{ij∗} ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}),

where 1 ≤ i, j ≤ r, and r + 1 ≤ α ≤ n.

Density for totally real r-planes about a fixed real q-plane.

The density for totally real planes L_{r} around a totally real plane L_{q} is given by

(2.2.3)  dL_{r[q]} = ⋀_{i≤j} ω_{ij∗} ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}),

where q + 1 ≤ i, j ≤ r, and r + 1 ≤ α ≤ n.

For q = 0 we get the density for totally real r-planes through the origin, which is the invariant volume element of the mixed Grassmann manifold

(2.2.4)  \frac{U(n)}{U(n − r) × O(r)}.

2.3.- Density for subspaces L_{2r+t}.

Under the reasonings of the preceding sections, this density corresponds to the homogeneous space M^{c}/H_{2r+t}, where H_{2r+t} is the subgroup of M^{c} of all motions that keep a subspace L_{2r+t}, determined by the unit vectors (e_{1}, …, e_{r+t}, e_{1∗}, …, e_{r∗}), invariant; which is, (2.3.1)

dL_{2r+t} = ⋀_{α}(ω_{α} ∧ ω_{α∗}) ⋀_{j} ω_{j∗} ⋀_{i≤j} ω_{ij∗} ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}) ⋀_{u,α}(ω_{uα} ∧ ω_{uα∗}) ⋀_{u,i}(ω_{ui} ∧ ω_{ui∗}),

where 1 ≤ u ≤ r, r + 1 ≤ i, j ≤ r + t, and r + t + 1 ≤ α ≤ n.

Density for linear spaces L_{2r+t} that leave a holomorphic 2q-plane fixed.

Let L_{2q}^{0} be a fixed holomorphic subspace with q < r. The density for linear spaces L_{2r+t} that leave L_{2q}^{0} invariant is

(2.3.2)  dL_{2r+t[2q]} = ⋀_{i≤j} ω_{ij∗} ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}) ⋀_{u,α}(ω_{uα} ∧ ω_{uα∗}) ⋀_{u,i}(ω_{ui} ∧ ω_{ui∗}),

where q + 1 ≤ u ≤ r, r + 1 ≤ i, j ≤ r + t, and r + t + 1 ≤ α ≤ n.

Density for linear spaces L_{2r+t} about a fixed totally real q-plane L_{q}^{0}.

We will distinguish two cases:

A) L_{q}^{0} is contained in the holomorphic section of L_{2r+t}; then, as in Note 2.2

(2.3.3.)  dL_{2r+t[q]} = dL_{2r+t[2q]}.

B) L_{q}^{0} is in the totally real section of L_{2r+t}, then

(2.3.4)  dL_{2r+t[q]} = ⋀_{i≤j} ω_{ij∗} ⋀_{i,α}(ω_{iα} ∧ ω_{iα∗}) ⋀_{u,α}(ω_{uα} ∧ ω_{uα∗}) ⋀_{u,i}(ω_{ui} ∧ ω_{ui∗}),

where 1 ≤ u ≤ r,  r + q + 1 ≤ i,  j ≤ r + t, and r + t + 1 ≤ α ≤ n.

In both cases, for q = 0, we have the density for the linear spaces L_{2r+t} through the origin, that is, the invariant volume element of the mixed Grassmannian

(2.3.5)  \frac{U(n)}{U(r) × U(n − r − t) × O(t)}.

Note 2.3. If r + t = n, we have the duality between the totally real r-planes and their orthogonal complement L_{2r+t},

(2.3.6)  dL_{2r+t[0]} = dL_{t[0]},

as follows immediately from (2.2.4) and Note 1.2.

The verification that all the forms defined in this section are densities is immediate using the structure equations (1.3) to verify that their exterior differentials vanish, which is a necessary and sufficient condition, [SA-3].

3.- Total curvatures of compact anti-invariant submanifolds in I C^{n}

The purpose of this section is to define the total curvatures of a compact totally real (anti-invariant) manifold M, in the euclidean space I C^{n}, and to express them as the integral over the manifold of the absolute value of certain curvatures defined in each point of M.

In [SA-1,2], we can find these curvatures, for compact manifolds immersed in the real euclidean space IR^{n}.

3.1.- The total absolute curvatures.

Let M be a compact, totally real C^{∞}-differentiable manifold of dimension q, without boundary, immersed in euclidean complex space I C^{n}.

In order to define the r-th total absolute curvature of M, we must distinguish several cases.

Case I.- We are going to consider that the tangent space at each point of M, either contains a totally real linear space, or it is included in a totally real linear space.

I.a) Case 1 ≤ r ≤ q.

Let L_{2n−r[0]} be a (2n − r)-dimensional linear space through the origin, with a holomorphic section of dimension 2(n − r) and a totally real section of dimension r.

Let Γ_{r} denotes the set of all totally real linear subspaces L_{r} in I C^{n} which are contained in some tangent space T_{p}M, passing through p and orthogonal to L_{2n−r[0]}.

Since M is compact the intersection Γ_{r} ∩ L_{2n−r[0]} will be a compact submanifold of L_{2n−r[0]}. Let μ(Γ_{r}∩L_{2n−r[0]}) be the measure of this manifold as submanifold of L_{2n−r[0]}. If the dimension of this submanifold is 0, then μ means the number of intersection points of Γ_{r} ∩ L_{2n−r[0]}.

3.1. Definition. We define the r-th total absolute curvature of M ⊂ I C^{n} as the mean value of the measures μ for all L_{2n−r[0]}; that is,

(3.1.1)  K_{r}(M) = \frac{∫ μ(Γ_{r} ∩ L_{2n−r[0]})dL_{2n−r[0]}}{∫ dL_{2n−r[0]}} ,

where the integration is over the mixed Grassmannian given by (2.2.4).

I.b) Case q < r ≤ n.

Using the same notations as above, denoting now Γ_{r} the set of all totally real r-planes which contain some T_{p}M and are orthogonal to L_{2n−r[0]}, the total absolute curvatures are defined by the same formulae (3.1.1).

Case II.- We are going to consider that the tangent space at each point of M is contained in a holomorphic linear space.

II.a) q ≤ r < n.

In this case Γ_{r} denotes all the holomorphic 2r-dimensional planes which contain some tangent space T_{p}M and are orthogonal to a holomorphic subspace L_{2(n−r)[0]}, and so the definition of K_{2r}(M) is the same as in (3.1.1), but integrating over the complex Grassmann manifold (2.1.4).

Case III.- We consider that the tangent space at each point of M is contained in a linear space L_{2s+t}.

III.a) q < s + t < n.

Now, Γ_{r} with r = 2s + t, denotes the set of all the planes L_{2s+t} that contain some T_{p}M an are orthogonal to a space of complementary dimension, and the total absolute curvatures are defined by (3.1.1) having to account that r = 2s + t.

3.2 Note. Since the measures are non negative, all the totally absolute curvatures are greater or equal than zero. We are interested only in the non-vanishing curvatures, which correspond to the cases that dimension of Γ_{r} ∩ L_{2n−r[0]} is non negative.

3.3 Proposition. The totally absolute curvatures K_{r}, defined in case I.a), are greater than zero if q = n, and r = 1.

In cases I.b), II.a) and III.a) all the curvatures K_{r} vanish.

Proof. We have seen in note 3.2 that K_{r} ≥ 0 in every case. We now proceed to compute the dimension of Γ_{r} ∩ L_{2n−r[0]}.

Since to each r-plane L_{r} orthogonal to L_{2n−r[0]} corresponds one and only one intersection point with L_{2n−r[0]}, the dimensions of Γ_{r} ∩L_{2n−r[0]} coincides with the dimension of Γ_{r} as submanifold of the set of totally real linear spaces in I C^{n}.

In order to obtain the dimension of Γ_{r} in case I.a), we consider the following manifolds;

Let G_{r} be the set of all totally real r-planes in I C^{n},then

(3.1.2)  dim(G_{r}) = \frac{1}{2}(r(r + 1)) + 2r(n − r) + r + 2(n − r).

Let H_{r} be the set of all r-planes L_{r} in I C^{n} which are contained in some T_{p}M passing through p. If we consider T_{p}M as a real linear space, all the r-planes through p in T_{p}M form the real Grassmann manifold G_{r,q} so the dimension of H_{r} is r(q − r) + q.

We denote by P_{r}, the totally real subspaces L_{r} that are orthogonal to L_{2n−r[0]}; then, dim(P_{r}) = 2n − r.

H_{r} and P_{r} are submanifolds of G_{r}; and Γ_{r} is the intersection of H_{r} and P_{r} consequently, the dimension of Γ_{r}, is

(3.1.3) dim(Γ_{r}) = 2n − r + r(q − r) + q − (\frac{1}{2}(r(r + 1)) + 2r(n − r) + r + 2(n − r)).

Simplifying this expression we get, dim(Γ_{r}) = \frac{1}{2}r^{2} − \frac{1}{2}r + rq + q + 2rn, which is zero if q = n, r = 1 and negative for the rest of values of r, q and n.

Hence, in order that K_{r} ≠ 0, it is necessary and sufficient that q = n, r = 1.

Now, we will study the dimension of Γ_{r} in the other cases. In case I.b), we will consider G_{r} and P_{r} as in the preceding case, and H_{r} will be the set of all r-planes L_{r} in I C^{n} which contain a fixed T_{p}M as q-linear space, then, the dimension of H_{r} is the degree of the form (2.2.3), i.e.

2(r − q)(n − r) + \frac{1}{2}(r − q)(r − q − 1) + q,

so the dimension of Γ_{r} is

(3.1.4) 2n−r+2(r−q)(n−r)+\frac{1}{2}(r−q)(r−q−1)+q−(\frac{1}{2}(r(r+1))+2r(n−r)+r+2(n−r)),

which is greater or equal than zero if and only if 2r − 4n + q + 1 ≥ 0, but this is not possible when one takes into account that q < r ≤ n.

For case II.a), the submanifolds G_{r}, P_{r} and H_{r} are the same as in case I.b), but using holomorphic planes L_{2r}, so we are left with dim(G_{r}) = 2(r +1)(n−r), dim(P_{r}) = 2(n − r), dim(H_{r}) = 2(r − q)(n − r) + q and we get

(3.1.5)  dim(Γ_{r}) = 2(r − q)(n − r) + q + 2n − 2r − (2r(n − r) + 2(n − r)),

which is non-negative if 2r − 2n + 1 ≥ 0, but this is impossible for q ≤ r < n.

Finally, using similar procedures in case III.a) as in the preceding cases we get that dim(Γ_{r}) is negative if q < s + t < n. □

Expression of the total absolute curvature K_{1}(M).

From definition (3.1.1) and proposition (3.3), and using the equivalence (2.3.6), we deduce that only one total absolute curvature doesn’t vanish; and this is

(3.1.6)  K_{1}(M) = \frac{∫_{G} μdL_{1[0]}}{∫_{G} dL_{1[0]}} ;  where  G = \frac{U(n)}{U(n − 1) × O(1)},

and Γ_{1} is the set of real lines in I C^{n}, which are contained in some tangent space T_{p}(M), passing through p and orthogonal to L_{2n−1[0]}; and μ is the number of intersection points of Γ_{1} with L_{2n−1[0]}.

We now compute the volume of G, in order to simplify this expression, i.e.

(3.1.7)  V ol(G) = ∫_{G} dL_{1[0]} = \frac{V ol(U(n))}{V ol(U(n − 1)) × V ol(O(1))}.

Let (e_{1}, …, e_{n}, e_{1∗}, …, e_{n∗}) be a unitary frame in IR^{2n}, the volume element of U(n) is given by

(3.1.8)  Ω = ⋀_{1≤i<j≤n} ω_{ij} ∧ ω_{ij∗} ⋀_{1≤k≤n} ω_{kk∗}.

We have the following relation between Ω and the density for U(n), obtained in [SA-4], Ω^{c}:

(3.1.9)  Ω^{c} = (\sqrt{−1})^{\frac{n(n+1)}{2}} 2^{\frac{n(n−1)}{2}} Ω.

Taking this into account, we find that the volume of U(n) with respect to the form Ω is

(3.1.10)  V ol(U(n)) = ∫_{G} Ω = 2^{n} ∏_{r=1}^{n} \frac{π^{r}}{(r − 1)!}.

On the other hand, V ol(O(1)) = 1 if the orientation is considered.

Then, we can express the total curvature K_{1}(M) as

(3.1.11)  K_{1}(M) = \frac{(n − 1)!}{2π^{n}} ∫_{G} μdL_{1[0]}.

3.2.- Local representation of K_{1}(M). Total (non absolute) curvatures.

Let (e_{1}, …, e_{n}, e_{1∗}, …, e_{n∗}) be a unitary frame in IR^{2n}, where (e_{1}, …, e_{n}) span the tangent space to M in the point x; then, the density for totally real lines through x is given by (2.2.4) which we will write now

dL_{1[x]} = ω_{1i∗} ∧ ω_{1j},  i = 1, …, n,  j = 2, …, n.

By rotation about the point x, the end point of the vector e_{1} moves on the unit sphere S^{2n−1} centered at x.

The scalar products de_{1}e_{j} = ω_{1j}, de_{1}e_{i∗} = ω_{1i∗}, are equal to the arc elements of S^{2n−1} at the end point of e_{1} in the direction of e_{j} and e_{i∗}, respectively; so that, denoting by dS^{2n−1} the (2n − 1)-dimensional volume element of S^{2n−1} at the end point of e_{1}, we have

dS^{2n−1} = ω_{1i∗} ∧ ω_{1j},  i = 1, …, n,  j = 2, …, n.

and K_{1}(M) is the mean value of μ over the unit sphere S^{2n−1}; which is the same expression that L. A. Santaló has obtained for total curvatures of n-dimensional submanifolds in IR^{2n}, [SA-2].

On the other hand, the assumption that (e_{1}, …, e_{n}) are tangents to M gives ω_{i∗} = 0. Taking this and formula (1.3) into account, and according to Cartan’s lemma, we have

(3.2.1)  ω_{ij∗} = h_{ik}^{j∗} ω_{k},  h_{ik}^{j∗} = h_{ki}^{j∗},

where h_{ik}^{j∗} are the coefficients of the second fundamental form with respect to e_{j∗}, which satisfies,

(3.2.2)  h_{jk}^{i} = h_{ik}^{j} = h_{ij}^{k};

where we write h_{ik}^{j∗} as h_{ik}^{j} following [K-M].

Moreover,

(3.2.3)  R_{ijhk} = R_{i∗j∗hk} = ∑_{r=1}^{n} (h_{ik}^{r} h_{jh}^{r} − h_{ih}^{r} h_{jk}^{r}) .

The local representation of K_{1}(M) will be, as in the real case, the integral over the manifold M of

(3.2.4)  Q_{1}(x) = \frac{(n − 1)!}{2π^{n}} ∫_{G_{1,n}} |f(x, e_{1})|dL_{1[x]},

where f(x, e_{1}) is, using (3.2.2), the determinant of the symmetric matrix

( \begin{matrix}h_{11}^{1} & h_{12}^{1} & … & h_{1n}^{1} \\ h_{12}^{1} & h_{22}^{1} & … & h_{2n}^{1} \\ ⋮ & ⋮ & ... & ⋮ \\ h_{1n}^{1} & h_{2n}^{1} & … & h_{nn}^{1} \end{matrix} ) .

The total (no absolute) curvature K_{1}^{∗}(M) is defined as the integral over the manifold M of

(3.2.5)  Q_{1}^{∗}(x) = \frac{(n − 1)!}{2π^{n}} ∫_{G_{1,n}} f(x, e_{1}) dL_{1[x]}.

By the same reasoning as in [SA-2], and using (3.2.3), we obtain that

(3.2.6) Q_{1}^{∗}(x) = \frac{1}{2^{n} (2π)^{n/2} (n/2)!}δ_{j_{1}j_{2}...j_{n}}^{i_{1}i_{2}...i_{n}}R_{i_{1}i_{2}j_{1}j_{2}} … R_{i_{n−1}i_{n}j_{n−1}j_{n}}  (n even),

where δ_{j_{1}j_{2}...j_{n}}^{i_{1}i_{2}...i_{n}} is equal to +1 or −1 according as (i_{1}i_{2} … i_{n}) is an even or odd permutation of (j_{1}j_{2} … j_{n}) and is otherwise zero. The summation of all i_{1}, i_{2}, … i_{n} and j_{1}, j_{2}, … j_{n} is from 1 to n.

If n is odd, then Q_{1}^{∗}(x) = 0.

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Departament de Matemàtiques i Informàtica, Universitat Jaume I, 12071 CASTELLON, SPAIN. E-mail: GUAL@ GREGAL.UJI.ES

Received 27 August 1992