On the Geometric Meaning of the Classical Equation of Gauss

Giovanni Battista Rizza

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

Let M be a submanifold of a Riemannian manifold M̃. Some geometrical relations, expressing the difference of the sectional (and bisectional) curvatures of M̃ and of M are obtained. These relations result to be equivalent to the classical Equation of Gauss.

The special case when M is λ-isotropic at a point x is also examined.

Mathematics Subject Classification: 53B25 Key words: Riemannian submanifolds, sectional and bisectional curvature

1 Introduction

The aim of the present paper is to obtain formulas, involving only geometrical elements, that result to be equivalent to the classical Equation of Gauss.

Let M̃ be a Riemannian manifold and M a submanifold of M. Consider a point x of M ⊂ M̃ and a pair p, q of oriented planes of T_{x}(M) ⊂ T_{x}(M̃).

It is known that, starting from the Equation of Gauss, we can derive a formula expressing the difference χ̃_{pq} − χ_{pq} of the bisectional curvatures of M̃ and of M. This relation, however, cannot be considered as completely satisfactory from a geometrical point of view (Sec.3).

In Sec. 4-7 we show how the above formula can be rewritten, in two different ways, in terms of the planes p, q only, by introducing convenient means on p, q (Theorem 1, Sec. 4; Theorem 2, Sec.6). In particular, when q = p we obtain two relations for the difference K̃_{p} − K_{p} of the sectional curvatures.

The research ends by showing that, if at any point x of M and for any plane p of T_{x}(M) one of the mentioned relations expressing K̃_{p} − K_{p} is satisfied, then we are able to derive the classical Equation of Gauss (Theorem 3, Sec.8).

In Sec. 9-13 we apply the general results of Sec. 4,6 to the special case, when the submanifold M is assumed to be λ-isotropic at the point x in the sense of B. O’Neill (Sec.11). Using the geometric notions of related bases and of canonical isometries (Sec.10), we obtain interesting formulas concerning sectional and bisectional curvatures of M and M̃ (Theorem 4, Sec.12). In particular, if M is umbilical at x, we are led to known relations.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 79-90.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

2 Preliminaries

Let V be an n-dimensional real vector space and g an inner product on V. In the sequel the 2-dimensional subspaces of V are called planes. Let p, q be two oriented planes of V.

We denote by ρ_{p}, ρ_{q} the rotations of \frac{π}{2} on p, q, respectively. Let i : p ⟶ q be an isometry. We say that i preserves the orientation if and only if i maps an oriented orthonormal basis of p onto an oriented orthonormal basis of q.

More explicitly, let X, Y be an oriented orthonormal basis of p, then ρ_{p} is the isomorphism defined by ρ_{p}X = Y, ρ_{p}Y = −X. Similarly for ρ_{q}. An isometry i preserves the orientation, if and only if iX, iY is an oriented orthonormal basis of q.

It is worth remarking that the geometrical notion of rotation of \frac{π}{2} in an oriented plane and of isometry preserving the orientation for a pair of oriented planes are intrinsic notions. In effect, we can easily prove that the definitions do not depend on the oriented orthonormal basis X, Y of the plane p.

3 The Equation of Gauss

Let M̃ = M̃(g) be an m̃-dimensional Riemannian manifold and M an m-dimensional submanifold (m ≥ 2), with induced metric still denoted by g.

We refer to [4]II Ch.7, to [1] Ch.2 and to [10] Ch.2 for the basic facts about the geometry of the submanifolds. In the sequel B denotes the second fundamental form and H = \frac{1}{m} trace B the mean curvature vector field of M.

Let x be a point of M ⊂ M̃ and R, R̃ be the Riemann curvature tensor of M, M̃ at x, respectively. Then, for any vectors X, Y, Z, W of T_{x}(M) ⊂ T_{x}(M̃), we have the well known Equation of Gauss

(1) R̃(X, Y, Z, W) − R(X, Y, Z, W) = g(B(X, W), B(Y, Z)) − g(B(X, Z), B(Y, W))

where the metric tensor g and the form B must be considered at x.

Now, let p, q be two oriented planes of T_{x}(M) ⊂ T_{x}(M̃). Denote by χ_{pq} and by χ̃_{pq} the bisectional curvatures of M and of M̃ with respect to the pair p, q. Let X, Y and Z, W be oriented orthonormal bases of p and of q, respectively. Since we have

χ_{pq} = R(X, Y, Z, W)  χ̃_{pq} = R̃(X, Y, Z, W)

(see for example [7],(1) p.148), from relation (1) we derive relation

(2)  χ̃_{pq} − χ_{pq} = g(B(X, W), B(Y, Z)) − g(B(X, Z), B(Y, W))

In particular, when q = p, we get the relation

(2′)  K̃_{p} − K_{p} = g(B(X, Y), B(X, Y)) − g(B(X, X), B(Y, Y))

concerning the sectional curvatures.

It is worth remarking that equation (2), (2′) already evidence the strict connection of the Equation of Gauss with the geometrical notions of bisectional and sectional creature. However, these equations cannot be considered as completely satisfactory from a geometrical point of view. As a matter of fact, the first members of (2),(2′) depend only on the oriented planes p, q; at the same time we see that in their second members oriented orthonormal bases occur.

In the sections 4-7, our problem will be that of rewriting the second members of (2), (2′) in terms of the oriented planes p,q only.

4 A first result

The problem posed at the end of Sec. 3 can be solved in more ways. The basic idea is that of making use of the notion of mean.

Let X, Y and Z, W be oriented orthonormal bases of the oriented planes p and q, respectively. Consider the sets

S_{p} = {P ∈ p | g(P, P) = 1} S_{q} = {Q ∈ q|g(Q, Q) = 1}.

We can write

P = X cos ϕ + Y sin ϕ Q = Z cos ψ + W sin ψ

and consequently

ρ_{p}P = Y cos ϕ − X sin ϕ ρ_{q}Q = W cos ψ − Z sin ψ

where ρ_{p}, ρ_{q} are the rotations of \frac{π}{2} on the oriented planes p, q, respectively (Sec. 2). Last we introduce the mean

(3)  m_{ρ} = \frac{1}{4π^{2}} ∫_{S_{p}×S_{q}} g(B(P, Q), B(ρ_{p}P, ρ_{q}Q)) dϕ dψ.

In particular, if q = p, we can choose Z = X, W = Y. Putting P = P_{1}, ϕ = ϕ_{1}, Q = P_{2}, ψ = ϕ_{2}, the mean m_{ρ} reduces to the mean

(3^{∗})  m_{ρ}^{∗} = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, P_{2}), B(ρ_{p}P_{1}, ρ_{p}P_{2})) dϕ_{1} dϕ_{2}.

We are now able to state Theorem l. The relations (2), (2′) of Sec. 3, derived from the Equation of Gauss, can be written in the form

(4)  χ̃_{pq} − χ_{pq} = −2 m_{ρ}

(4′)  K̃_{p} − K_{p} = −2 m_{ρ}^{∗}

where m_{ρ}, m_{ρ}^{∗} are given by (3), (3^{∗}).

Since the notion of rotation of \frac{π}{2} in an oriented plane is an intrinsic notion (Sec.2), Theorem 1 gives an exaustive answer to our problem.

5 Proof of Theorem 1

Since we have

B(P, Q) = cos ϕ cos ψ B(X, Z) + cos ϕ sin ψ B(X, W)
+ sin ϕ cos ψ B(Y, Z) + sin ϕ sin ψ B(Y, W)
B(ρ_{p}P, ρ_{p}Q) = cos ϕ cos ψ B(Y, W) − cos ϕ sin ψ B(Y, Z)
− sin ϕ cos ψ B(X, W) + sin ϕ sin ψ B(X, Z)

we derive

g(B(P, Q), B(ρ_{p}P, ρ_{q}Q)) = g(B(X, Z), B(Y, W))[cos^{2} ϕ cos^{2} ψ + sin^{2} ϕ sin^{2} ψ]
− g(B(X, W), B(Y, Z))[cos^{2} ϕ sin^{2} ψ + sin^{2} ϕ cos^{2} ψ] + ...

Here and in the sequel, the dots stand for terms, that will give zero by integration on S_{p} or on S_{q}.

It is elementary to check that the second member of relation (2) of Sec.3 can be replaced by −2 m_{ρ} and this proves Theorem l.

6 A second result

In the present section, by using the intrinsic notion of isometry preserving the orientation (Sec.2), we will be able to give a second answer to the problem, we posed at the end of Sec.3.

We begin by remarking that to any oriented plane p of T_{x}(M) ⊂ T_{x}(M̃) we can intrinsecally associate the vector

(5)  B_{p} = \frac{1}{2π} ∫_{S_{p}} B(P, P)dϕ

where S_{p} is the set of the unit vectors of p. In other words, we consider the mean of the normal curvature vectors ([5], p.149) of the unit vectors of p.

Now, let P_{1}, P_{2} be two vectors of S_{p} and let i : p ⟶ q be any isometry, that preserves the orientation (Sec.2). We introduce the means

(6)  m_{C}(i) = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, iP_{2}), B(P_{2}, iP_{1}))dϕ_{1}dϕ_{2}

(7)  m_{S}(i) = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, iP_{1}), B(P_{2}, iP_{2}))dϕ_{1}dϕ_{2}

Remark that in (7) the vectors P_{1}, P_{2} act separately, while in (6) they act in a crossed way. Hence the notations m_{S}(i), m_{C}(i), respectively.

In particular, if q = p, we can choose i = identity. Then m_{C}(i), m_{S}(i) reduce to

(6^{∗})  m_{N} = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, P_{2}), B(P_{1}, P_{2}))dϕ_{1}dϕ_{2}

(7^{∗})  g(B_{p}, B_{p}) = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, P_{1}), B(P_{2}, P_{2}))dϕ_{1}dϕ_{2}

respectively.

Note that the second member of (6*) is, essentially, the integral of a norm. Hence the notation m_{N}. To prove (7*), just remark that the second member can be written in the form

g(\frac{1}{2π} ∫_{S_{p}} B(P_{1}, P_{1})dϕ_{1}, \frac{1}{2π} ∫_{S_{p}} B(P_{2}, P_{2})dϕ_{2})

and that B_{p} is defined by (5).

We are now able to state Theorem 2. The relations (2),(2′) of Sec. 3, derived from the Equation of Gauss, can be written in the form

(8)  χ̃_{pq} − χ_{pq} = 2(m_{C}(i) − m_{S}(i))

(8′)  K̃_{p} − K_{p} = 2(m_{N} − g(B_{p}, B_{p}))

where i : p ⟶ q is any isometry preserving the orientation and m_{C}(i), m_{S}(i), m_{N}, B_{p} are defined by (6),(7),(6*),(5), respectively.

Since the notions of isometry preserving the orientation, that occurs in (6),(7), as well as the definitions of m_{N} and of B_{p}, are intrinsic, so Theorem 2 represents a completely satisfactory solution of the problem at the end of Sec.3.

7 Proof of Theorem 2

Let X, Y be an oriented orthonormal basis of the oriented plane p. Consequently, since i : p ⟶ q is assumed to be an isometry that preserves the orientation, we can choose as oriented orthonormal basis of q the pair Z = iX, W = iY (Sec.2).

Consequently, if P_{1}, P_{2} are vectors of S_{p}, i.e.

P_{1} = X cos ϕ_{1} + Y sin ϕ_{1} P_{2} = X cos ϕ_{2} + Y sin ϕ_{2}
iP_{1} = Z cos ϕ_{1} + W sin ϕ_{1} iP_{2} = Z cos ϕ_{2} + W sin ϕ_{2}.

It follows

B(P_{1}, iP_{2}) = B(X, Z) cos ϕ_{1} cos ϕ_{2} + B(X, W) cos ϕ_{1} sin ϕ_{2}
+ B(Y, Z) sin ϕ_{1} cos ϕ_{2} + B(Y, W) sin ϕ_{1} sin ϕ_{2}
B(P_{2}, iP_{1}) = B(X, Z) cos ϕ_{1} cos ϕ_{2} + B(X, W) sin ϕ_{1} cos ϕ_{2}
+ B(Y, Z) cos ϕ_{1} sin ϕ_{2} + B(Y, W) sin ϕ_{1} sin ϕ_{2}
B(P_{1}, iP_{1}) = B(X, Z) cos^{2} ϕ_{1} + B(X, W) cos ϕ_{1} sin ϕ_{1}
+ B(Y, Z) sin ϕ_{1} cos ϕ_{1} + B(Y, W) sin^{2} ϕ_{1}
B(P_{2}, iP_{2}) = B(X, Z) cos^{2} ϕ_{2} + B(X, W) cos ϕ_{2} sin ϕ_{2}
+ B(Y, Z) sin ϕ_{2} cos ϕ_{2} + B(Y, W) sin^{2} ϕ_{2}.

Therefore we can write

g(B(P_{1}, iP_{2}), B(P_{2}, iP_{1})) =
g(B(X, Z), B(X, Z)) cos^{2} ϕ_{1} cos^{2} ϕ_{2} + g(B(Y, W), B(Y, W)) sin^{2} ϕ_{1} sin^{2} ϕ_{2}
+g(B(X, W), B(Y, Z))[sin^{2} ϕ_{1} cos^{2} ϕ_{2} + cos^{2} ϕ_{1} sin^{2} ϕ_{2}] + ...

g(B(P_{1}, iP_{1}), B(P_{2}, iP_{2})) =
g(B(X, Z), B(X, Z)) cos^{2} ϕ_{1} cos^{2} ϕ_{2} + g(B(Y, W), B(Y, W)) sin^{2} ϕ_{1} sin^{2} ϕ_{2}
+g(B(X, Z), B(Y, W))[sin^{2} ϕ_{1} cos^{2} ϕ_{2} + cos^{2} ϕ_{1} sin^{2} ϕ_{2}] + ...

where the dots stand for terms, that will give zero by integration on S_{p} × S_{p}

It is not difficult now to see that the second member of relation (2) of Sec.3 can be replaced by 2(m_{C}(i) − m_{S}(i)) and this proves Theorem 2.

8 The main result

The aim of the present section is to prove Theorem 3. At any point x of the submanifold M the classical Equation of Gauss results to be equivalent to relation (4′) as well as to relation (8′) for any plane p of T_{x}(M). Corollary l. Each one of the relations (4′) of Sec.4, (8′) of Sec.6, both concerning the sectional curvature, summarizes the whole geometrical content of the Equation of Gauss.

We begin with Corollary 2. For the means m_{ρ}, m_{ρ}^{∗} and the means m_{C}(i), m_{S}(i), m_{N}, B_{p}, introduced in Sec. 4, 6, we have

m_{ρ} = m_{S}(i) − m_{C}(i) m_{ρ}^{∗} = g(B_{p}, B_{p}) − m_{N}.

This fact follows immediately by comparing (4), (4′) with (8), (8′), respectively. As a consequence, we see that relations (4′) and (8′), occurring in Theorem 3, are

equivalent. On the other hand, the last sentence of Sec.4 implies that (4) and (2) are equivalent. In particular (4′) results to be equivalent to relation (2′).

In conclusion, since (2),(2′) have been derived from the Equation of Gauss, to prove Theorem 3 we have only to prove (1) for any \overline{X}, \overline{Y} , \overline{Z}, \overline{W} of T_{x}(M), starting from (2′) for any oriented plane p of T_{x}(M), being X, Y any oriented orthonormal basis of p.

Recalling the definition of sectional curvature ([4]I, p.202), we first rewrite (2′) as

(1′) R̃(X, Y, X, Y) − R(X, Y, X, Y) = g(B(X, Y), B(X, Y)) − g(B(X, X), B(Y, Y))

where X, Y is any pair of orthonormal vector of T_{x}(M). Then, it is elementary to check that (1′) holds true for any pair \overline{X}, \overline{Y} of vectors of T_{x}(M).

Consider now the quadrilinear forms

Q_{1}(\overline{X}, \overline{Y} , \overline{Z}, \overline{W}) = R̃(\overline{X}, \overline{Y} , \overline{Z}, \overline{W}) − R(\overline{X}, \overline{Y} , \overline{Z}, \overline{W})
Q_{2}(\overline{X}, \overline{Y} , \overline{Z}, \overline{W}) = g(B(\overline{X}, \overline{W}), B(\overline{Y} , \overline{Z})) − g(B(\overline{X}, \overline{Z}), B(\overline{Y} , \overline{W}))

It is an easy matter to check that Q_{1} and Q_{2} satisfy the conditions a, b, c of p.198 of [4]I,

Finally, since by a preceeding remark we have Q_{1}(\overline{X}, \overline{Y} , \overline{X}, \overline{Y}) = Q_{2}(\overline{X}, \overline{Y} , \overline{X}, \overline{Y}), by Proposition l.2 of [4]I, p.198 we find Q_{1} = Q_{2}, that is the Equation of Gauss.

Therefore the proof of Theorem 3 is complete.

9 The means \overline{m}_{C}(i), \overline{m}_{S}(i)

In order to treat the special case of Sec.12 we need some premises.

First of all, we introduce the means

(\overline{6})  \overline{m}_{C}(i) = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, P_{2}), B(iP_{1}, iP_{2}))dϕ_{1}dϕ_{2}

(\overline{7})  \overline{m}_{S}(i) = \frac{1}{4π^{2}} ∫_{S_{p}×S_{p}} g(B(P_{1}, P_{1}), B(iP_{2}, iP_{2}))dϕ_{1}dϕ_{2}

that are analogous to the means m_{C}(i), m_{S}(i) defined by (6),(7) in Sec. 6. Proceeding as in Sec.7, we get

g(B(P_{1}, P_{2}), B(iP_{1}, iP_{2})) =

g(B(X, X), B(Z, Z)) cos^{2} ϕ_{1} cos^{2} ϕ_{2} + g(B(Y, Y), B(W, W)) sin^{2} ϕ_{1} sin^{2} ϕ_{2}

+g(B(X, Y), B(Z, W))[cos^{2} ϕ_{1} sin^{2} ϕ_{2} + sin^{2} ϕ_{1} cos^{2} ϕ_{2}] + ...

g(B(P_{1}, P_{1}), B(iP_{2}, iP_{2})) =

g(B(X, X), B(Z, Z)) cos^{2} ϕ_{1} cos^{2} ϕ_{2} + g(B(Y, Y), B(W, W)) sin^{2} ϕ_{1} sin^{2} ϕ_{2}

+g(B(X, X), B(W, W)) cos^{2} ϕ_{1} sin^{2} ϕ_{2} + g(B(Y, Y), B(Z, Z)) sin^{2} ϕ_{1} cos^{2} ϕ_{2} + ...

where the dots stand for terms, that will give zero by integration on S_{p} × S_{p}.

Now, taking into account these relations and the analogous ones of Sec.7 concerning g(B(P_{1}, iP_{2}), B(P_{2}, iP_{1})) and g(B(P_{1}, iP_{1}), B(P_{2}, iP_{2})) we can write the relations

(10)  2(m_{C}(i) + \overline{m}_{C}(i) + m_{S}(i)) = E(i)

(11)  \overline{m}_{S} = g(B_{p}, B_{q})

where

2E(i) = g(B(X, X), B(Z, Z)) + 2g(B(X, Z), B(X, Z))
(12) + g(B(Y, Y), B(W, W)) + 2g(B(Y, W), B(Y, W))
+ 2[g(B(X, Y), B(Z, W)) + g(B(X, Z), B(W, Y)) + g(B(X, W), B(Y, Z))].

To prove (11) it is worth remarking that from definition (5) of Sec.6 we can derive

(13)  B_{p} = \frac{1}{2}(B(X, X) + B(Y, Y)) B_{q} = \frac{1}{2}(B(Z, Z) + B(W, W)).

10 Canonical isometries

We recall first that two oriented orthonormal bases X, Y and Z, W for the oriented planes p, q, respectively, are said to be related bases, if we have

(14)  g(X, W) = g(Y, Z) = 0

The geometric notion of related bases plays an essential role in the papers [8], [3]. We refer to [9] for notations and details.

Let X, Y and Z, W be a pair of related bases of p, q. Then the isometry i_{∗} : p ⟶ q defined by i_{∗}X = Z, i_{∗}Y = W is said to be a canonical isometry. By definition, canonical isometries preserve the orientation. Denote by −i_{∗} the isometry defined by (−i_{∗})P = −(i_{∗}P) for any vector P of p. Since X, Y and −Z, −W is a pair of related bases of p, q, also −i_{∗} is a canonical isometry. Consequently Proposition 1 of [9] ensures the existence of two canonical isometries for any pair of oriented planes.

Let’s denote by α_{m}, α_{M} the minimum, maximum value of the angle that a line (1-dimensional subspace) of p forms with the plane q. Referring to Remarks 1,2,3 of [9], we are now able to state Remark 1. If p, q are not isoclinic planes, i.e. α_{m} ≠ α_{M}, there exist only two canonical isometries. If p, q are isoclinic not strictly orthogonal planes, that is α_{m} = α_{M} ≠ \frac{π}{2} , we have two cases, according to the fact that we act on the bases X, Y and Z, W by equal or opposite rotations. Correspondingly, we have two or ∞^{1} canonical isometries. Finally, if p, q are strictly orthogonal, that is α_{m} = α_{M} = \frac{π}{2} , any isometry preserving the orientation is a canonical isometry; so we have ∞^{2} canonical isometries.

The proof of Remark 1 is elementary.

We end the section by recalling that by virtue of (14) we have ([7],(4))

(15)  cos pq = g(X, Z) g(Y, W).

11 λ-isotropy

In 1965 B. O’Neill introduced and studied the λ-isotropy of the submanifolds ([5][6]). A submanifold M of M̃ is said to be λ-isotropic at x(λ ≥ 0), if we have

(16)  g(B(X, X), B(X, X)) = λ^{2}(g(X, X))^{2}

for any vector X of T_{x}(M).

It is immediate to check that the above definition is equivalent to the original one of B. O’Neill. Moreover we have Proposition l. The submanifold M of M̃ results to be λ-isotropic (λ ≥ 0) at x, if and only if we have

(17)  \begin{matrix}g(B(X, Y), B(Z, W)) + g(B(X, Z), B(W, Y)) + g(B(X, W), B(Y, Z)) = \\ = λ^{2}[g(X, Y)g(Z, W) + g(X, Z)g(W, Y) + g(X, W)g(Y, Z)] \end{matrix}

for any X, Y, Z, W of T_{x}(M).

It is worth remarking that the relation of Lemma 1 as well as the relations (1),(2),(3) of Lemma 2 of [6] are special cases of (17).

Since relation (16) follows immediately from (17), to prove Proposition 1 we have only to show that, starting from (16), we can obtain relation (17). In effect, this can be done by using iterated polarizations and by remarking that at any step of the proof you can replace a vector variable, say X, by kX(k ∈ IR) and then use the identity principe of polynomials.

Finally, an immediate consequence of (17) is relation

(18)  \begin{matrix}g(B(X, X), B(Y, Y)) + 2g(B(X, Y), B(X, Y)) = \\ = λ^{2}[g(X, X)g(Y, Y) + 2g(X, Y)g(X, Y)] \end{matrix}

for any X, Y of T_{x}(M).

12 Special cases

We consider the case when the submanifold M of M̃ is λ-isotropic (λ ≥ 0) at the point x.

Taking account of the remarks of Sec. 9,10,11, we are now able to state Theorem 4. For any canonical isometry i_{∗} : p ⟶ q, we have

(19)  χ̃_{pq} − χ_{pq} = 4 m_{C}(i_{∗}) + 2 \overline{m}_{C}(i_{∗}) − E = −4 m_{S}(i_{∗}) − 2 \overline{m}_{C}(i_{∗}) + E

(20)  K̃_{p} − K_{p} = 2(3 m_{N} − 2λ^{2}) = 2λ^{2} − 3g(B_{p}, B_{p})

where

(21)  E = λ^{2}(1 + cos^{2} α_{m} + cos^{2} α_{M} + cos pq)

Corollary 3. For any plane p of T_{x}(M) we have

(22)  −4λ^{2} ≤ K̃_{p} − K_{p} ≤ 2λ^{2}

Few remarks complete the subject

Let’s denote by p′ the same plane as p with opposite orientation. Taking into account of (3) of [7], we get

(23)  χ̃_{pp′} − χ_{pp′} = 2(2λ^{2} − 3 m_{N}) = 3g(B_{p}, B_{p}) − 2λ^{2}.

Moreover, it is immediate to check that we have

m_{C}(−i^{∗}) = m_{C}(i^{∗}) m_{S}(−i^{∗}) = m_{S}(i^{∗}) \overline{m}_{C}(−i^{∗}) = \overline{m}_{C}(i^{∗}).

So, by virtue of Remark 1 of Sec.10, if we have α_{m} ≠ α_{M} or α_{m} = α_{M} ≠ \frac{π}{2} (first case), then these three means are invariant under changements of the canonical isometries.

Last, the expression E depends only on the geometry of the pair of oriented planes p, q. When p and q are orthogonal or isoclinic or have a line in common, then E takes very simple forms.

We end the paper by considering the case when the submanifold M is umbilical at the point x.

Since we have B(X, Y) = g(X, Y)H for any X, Y of T_{x}(M), equation (16) is satisfied and M is λ-isotropic with λ = |H|. On the other hand, it is elementary to prove that, in the present case, we have

4 m_{C}(i^{∗}) = g(H, H)[cos^{2} α_{m} + cos^{2} α_{M}],  2 m_{C}(i^{∗}) = g(H, H)
4 m_{S}(i^{∗}) = g(H, H)[cos^{2} α_{m} + cos^{2} α_{M} + cos pq],  m_{N} = g(H, H)
E = g(H, H)[1 + cos^{2} α_{m} + cos^{2} α_{M} + cos pq],  B_{p} = H

Consequently (19), (20) reduce to known relations (Cf. (8) of [2]).

13 Proofs

To prove Theorem 4 , we consider a pair X, Y and Z, W of related bases defining the canonical isometry i^{∗}. Taking account of (17), (18) and of (14), from relation (12) we get

E(i^{∗}) = λ^{2}(1 + (g(X, Z))^{2} + (g(Y, W))^{2} + g(X, Z)g(Y, W))

Then, using (11) of [9] and (15), we find that E(i^{∗}) coincide with the expression E defined by (21). Finally, starting from (8),(10) with i = i^{∗}, we prove (19) by sum and difference.

When q = p we can choose Z = X, W = Y. Remarking that X, Y and Z, W are related bases, we find that the corresponding canonical isometry is i^{∗} = identity. We know from Sec.6 that m_{C}(i), m_{S}(i) reduce to m_{N}, g(B_{p}, B_{p}), respectively. It is immediate that also \overline{m}_{C}(i) reduces to M_{N}. On the other hand, since q = p implies cos pq = 1 and α_{m} = α_{M} = 0, the expression E reduces to 4λ^{2}. Now, taking into account that relation (10) gives

(24)  2m_{N} + g(B_{p}, B_{p}) = 2λ^{2}

starting from (19) we prove (20).

This completes the proof of Theorem 4.

Finally, Corollary 3 is an immediate consequence of (20).

References

[1] B.-Y. Chen, Geometry of submanifolds, Dekker, New York 1973.

[2] S. Ianus and G. B. Rizza, Some submanifolds of a parakähler manifold, Rend. Circ. Mat. Palermo 47 (1998), 71-80.

[3] S. Ianus and G.B. Rizza, On sectional and bisectional curvature of the H-umbilical submanifolds, Internat. J. Math. Math. Sci. (2002).

[4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I and II, Interscience, New York, 1963 and 1969.

[5] B. O’Neill, Umbilics of constant curvature immersions, Duke Math. J. 32 (1965), 149-159.

[6] B. O’Neill, Isotropic and Kähler immersions, Canadian J. Math. 17 (1965), 907-915.

[7] G.B. Rizza, On the bisectional curvature of a Riemannian manifold, Simon Stevin 61 (1987), 147-155.

[8] G.B. Rizza, On almost Hermitian manifolds with constant holomorphic curvature at a point, Tensor 50 (1991), 79-89.

[9] G.B. Rizza, On the geometry of a pair of oriented planes, Riv. Mat. Univ. Parma 4 (2001), 217-228.

[10] K. Yano and M. Kon, Anti-invariant submanifolds, Dekker, New York 1976.

Giovanni Battista Rizza

Dip. Mat. Univ. Parma

Via D’Azeglio, 85

43100 Parma, Italia

email: rizza@prmat.math.unipr.it