Mean Curvature Comparison for Tubular Hypersurfaces in Symmetric Spaces

X. Gual-Arnau and R. Masó

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

Abstract We obtain comparison results for the mean curvature of tubular hypersurfaces, P_{t}, around a submanifold P of a riemannian manifold M, with bounded curvature, taking as a model tubular hypersurfaces around totally geodesic, curvature preserving submanifolds in symmetric spaces of arbitrary rank, and we give an application to get estimates for the relative volume.

Mathematics Subject Classification: 53C21, 53C35. Key words: comparison theorems submanifolds, mean curvature, symmetric spaces, totally geodesic, tubes.

1 Introduction

The first purpose of this paper is to obtain comparison theorems for the mean curvature of tubular hypersurfaces, P_{t}, around a submanifold P of a Riemannian manifold M, with bounded curvature, taking as a model tubular hypersurfaces around totally geodesic, curvature preserving submanifolds in symmetric spaces of arbitrary rank. Results of this type have been widely studied in the literature considering different rank-one symmetric spaces as a model. Moreover, these results have been applied to obtain comparison results for geometric Riemannian invariants such as volume, mean exit time,.... (see, for instance, [10], [7], [4], [5], [12], [13], [11]).

The bounds imposed to the q−mean curvatures defined in [1, page 253] to obtain the comparison theorems are given from the restricted roots of the symmetric space. That is the reason because these bounds are constant for rank-one symmetric spaces, but they depend, in general, on the vector used to define the q−mean curvatures in arbitrary rank symmetric spaces. The above situation is closely related with the fact that the eigenvalues of the Weingarten map, S(t), of a tubular hypersurface in a symmetric space, are constant for rank-one symmetric spaces but depend on the vector used to define S(t) for arbitrary rank symmetric spaces ([8], [14]).

The second purpose of the paper is to obtain a comparison result for the relative volume using, as a model, totally geodesic submanifolds in the symmetric space for which the first conjugate locus and the cut-focal locus agree ([2]).

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 53-62.

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

We will consider along the paper compact symmetric spaces, however using the duality between compact and noncompact symmetric spaces we will extend the results to noncompact symmetric spaces in the Appendix at the end of the paper.

2 Preliminaries

Let M be a Riemannian manifold of dimension n. Let P be a submanifold of M of dimension s. We shall denote by N P the normal bundle of P in M and by N_{p}P the fibre of N P over p ∈ P. Given a unitary vector u ∈ N_{p}P, we consider an orthogonal decomposition

T_{p}M = (⨁_{j=1}^{l_{1}} H_{j}) ⊕ (⨁_{i=1}^{l_{2}} V_{i}) , (2.1)

where

T_{p}P = ⨁_{j=1}^{l_{1}} H_{j} and (T_{p}P)^{⊥} = ⨁_{i=1}^{l_{2}} V_{i}

and such that u ∈ V_{1}, dim V_{1} = r, dim V_{i} = m_{i} (i = 2, …, l_{2}), dim H_{1} = q − r and dim H_{j} = n_{j} (j = 2, …, l_{1}).

The m_{i}-Ricci curvature K(u, V_{i}) of u at V_{i} (called the m_{i}-mean curvature in [1, page 253]) is defined as

K(u, V_{i}) = ∑_{k_{2}=1}^{m_{i}} R(u, X_{i_{k_{2}}}^{⊥} , u, X_{i_{k_{2}}}^{⊥}); (2.2)

where {X_{i_{k_{2}}}^{⊥} , k_{2} = 1, …, m_{i}} is an orthonormal basis of V_{i}. This definition can be extended to any subspace of T_{p}M and, in particular, to H_{j} (j = 1, …, l_{1}).

Let γ_{u}(t) be the geodesic such that γ_{u}(0) = p ∈ P and γ′_{u}(0) = u ∈ N_{√}P, and τ_{t}

the parallel transport along γ_{u}(t) from 0 to t. Then, if V_{i}^{t} = τ_{t}V_{i} and H_{j}^{t} = τ_{t}H_{j}, we have

T_{γ_{u}(t)}M = (⨁_{j=1}^{l_{1}} H_{j}^{t}) ⊕ (⨁_{i=1}^{l_{2}} V_{i}^{t}) .

Let P_{t} be the tubular hypersurface around P of radius t. We denote by S(t) the Weingarten map of P_{t} and we consider in P_{t} the operator R(t)X = R(γ′_{u}(t), X)γ′_{u}(t).

Lemma 1.1. [6]. Let A_{P} (t) denote the volume of P_{t}; then,

(2.3)  S′(t) = S^{2}(t) + R(t),
(2.4)  \frac{θ′_{u}(t)}{θ_{u}(t)} = − (\frac{n − s − 1}{t} + tr(S(t))) ,
(2.5)  A_{p}(t) = t^{n−s−1} ∫_{P} ∫_{S^{n−s−1}(1)} θ_{u}(t) du dP,

where θ_{u}(t) is the infinitesimal change of volume function in the direction of u and S^{n−s−1}(1) denotes the unit sphere in N_{p}P.

Now, if f(u) = inf{t > 0 / γ_{u}(t) is a conjugate point of p}, c(u) = sup{t > 0 / d(p, γ_{u}(t)) = t} (d being the distance function in M) and z(θ_{u}(t)) is the first positive zero of the function θ_{u}(t), we have

Lemma 1.2.

(2.6)  c(u) ≤ f(u) = z(θ_{u}(t)),
(2.7)  Vol(M) = ∫_{P} ∫_{S^{n−s−1}(1)} ∫_{0}^{c(u)} t^{n−s−1}θ_{u}(t) dt du dP.

To compare, in the next section, the trace of S(t), we will use a pair (M̃, P̃) as a model, where M̃ = G/K is a compact symmetric space of dimension n and P̃ is a totally geodesic submanifold of M̃ of dimension s, which satisfies the following properties.

Let g = k + m be the canonical decomposition of M̃ (m is indetified with the tangent space of M̃ at any point), and h a maximal abelian subspace of m. Since P̃ is totally geodesic, it is also a symmetric space P̃ = U/L. Let u = l + p be the canonical decomposition of P̃, where p is identified with the tangent space of P̃ at any point.

We assume that the orthogonal complement of p in m, p^{⊥}, is a Lie triple system, i.e. [p^{⊥}, [p^{⊥}, p^{⊥}]] ⊂ p^{⊥}, (2.8)

then, [9], P̃^{⊥} = Exp(p^{⊥}) is a totally geodesic submanifold of M̃, and P̃^{⊥} = U′/L′ is also a Riemannian globally symmetric space.

A list of pairs (P̃, P̃^{⊥}) for all compact symmetric spaces can be found in [3]. We say that P̃ is a totally geodesic curvature preserving submanifold of M̃, be-

cause, for each vector u ∈ p^{⊥}, the curvature operator R_{u} satisfies, from (2.8), the following condition of preserving the curvature,

R_{u}(p) ⊂ p and R_{u}(p^{⊥}) ⊂ p^{⊥}. (2.9)

Let a be a maximal abelian subspace of p^{⊥} with u ∈ a ⊂ h, then, if rank(P̃^{⊥}) = r and rank(M̃) = q, we have that r ≤ q. Let b = h ∩ p.

Let α_{i} (1 ≤ i ≤ l_{2}) be the positive restricted root system of p^{⊥} and β_{j} (1 ≤ j ≤ l_{1}) that of p. From (2.9), m can be decomposed as

m = (a ⊕ ∑_{i=2}^{l_{2}} m_{i}) ⊕ (b ⊕ ∑_{j=2}^{l_{1}} n_{j}) , (2.10)

where m_{i} is the root subspace of dimension m_{i} corresponding to α_{i}, (α_{1}(u) = 0, m_{1} = r), and n_{j} is the root subspace of dimension n_{j} corresponding to β_{j}, (β_{1}(u) = 0, n_{1} = q − r).

Let {X_{i_{k}}^{⊥}, k = 1, …, l_{2}} be an orthonormal basis of p^{⊥} such that X_{1_{1}}^{⊥} = u, {X_{1_{k}}^{⊥}, k = 1, …, r} ⊂ a and {X_{i_{k}}^{⊥}, k = 1, …, m_{i}} ⊂ m_{i}. Let {X_{j_{k}}^{⊤}, k = 1, …, l_{1}} be an orthonormal basis of p such that {X_{1_{k}}^{⊤}, k = 1, …, q − r} ⊂ b and {X_{j_{k}}^{⊤}, k = 1, …, n_{j}} ⊂ n_{j}.

In this way we get a basis {X_{j_{k_{1}}}^{⊤} , X_{i_{k_{2}}}^{⊥} }, (k_{1} = 1, …, n_{j}, j = 1, …, l_{1}), (k_{2} = 1, …, m_{i}, i = 1, …, l_{2}), of m which diagonalizes R_{u}, with eigenvalues given by

{β_{j}^{2}(u), α_{i}^{2}(u)}, (j = 1, …, l_{1}), (i = 1, …, l_{2}).

The m_{i}-Ricci curvature of u at m_{i} and the n_{j}-Ricci curvature of u at n_{j} are, respectively, K(u, m_{i}) = m_{i}α_{i}^{2}(u) and K(u, n_{j}) = n_{j}β_{j}^{2}(u). (2.11)

The Ricci curvature of M̃ with respect to u is

ρ(u, u) = ∑_{i=2}^{l_{2}} m_{i}α_{i}^{2}(u) + ∑_{j=2}^{l_{1}} n_{j}β_{j}^{2}(u). (2.12)

Let {E_{j_{k_{1}}}^{⊤} (t), E_{i_{k_{2}}}^{⊥} (t)}, (k_{1} = 1, …, n_{j}, j = 1, …, l_{1}), (k_{2} = 1, …, m_{i}, i = 1, …, l_{2}), be the parallel transport of {X_{j_{k_{1}}}^{⊤} , X_{i_{k_{2}}}^{⊥} } along the geodesic γ_{u}(t). The

operators S(t) and R(t) corresponding to M̃ satisfy:

\begin{aligned}R̃(t)E_{1_{k_{1}}}^{⊤} (t) & = & 0, k_{1} = 2, …, q − r, \\ R̃(t)E_{j_{k_{1}}}^{⊤} (t) & = & β_{j}^{2}(u)E_{j_{k_{1}}}^{⊤} (t), k_{1} = 1, …, n_{j}, j = 2, …, l_{1}, \\ R̃(t)E_{1_{k_{2}}}^{⊥} (t) & = & 0, k_{2} = 1, …, r, \\ R̃(t)E_{i_{k_{2}}}^{⊥} (t) & = & α_{i}^{2}(u)E_{i_{k_{2}}}^{⊥} (t), k_{2} = 1, …, m_{i}, i = 2, …, l_{2}, \end{aligned} (2.13)

and

\begin{aligned}S̃(t)E_{1_{k_{1}}}^{⊤} (t) & = & 0, k_{1} = 2, …, q − r, \\ S̃(t)E_{j_{k_{1}}}^{⊤} (t) & = & β_{j}(u) tan(tβ_{j}(u))E_{j_{k_{1}}}^{⊤} (t), k_{1} = 1, …, n_{j}, j = 2, …, l_{1}, \\ S̃(t)E_{1_{k_{2}}}^{⊥} (t) & = & −1/t E_{1_{k_{2}}}^{⊥} (t), k_{2} = 1, …, r, \\ S̃(t)E_{i_{k_{2}}}^{⊥} (t) & = & −α_{i}(u) cot(tα_{i}(u))E_{i_{k_{2}}}^{⊥} (t), k_{2} = 1, …, m_{i}, i = 2, …, l_{2}, \end{aligned} (2.14)

Moreover, when the first conjugate locus of P̃ and the cut-focal locus of P̃ agree (see examples in [2]), from (2.6), the minimal focal distance of P̃ in M̃, c(P̃) = min{c(u) / u ∈ N_{p}P̃}, is given by

c(P̃) = inf \{\frac{π}{2β_{j}(u)}, \frac{π}{α_{i}(u)} / u ∈ a and ∥u∥ = 1\} . (2.15)

In the following we will identify T_{p}M with T_{p′} M̃ and T_{p}P with T_{p′} P̃, and, given a unitary vector u ∈ V_{1} ⊂ N_{p}P, having in mind that p^{⊥} = ∪_{k}Ad(k)a, we can suppose that u ∈ a with restricted roots {α_{i}(u), β_{j}(u)}.

3 Mean curvature comparison

Now, we will obtain two comparison theorems when the pair (M̃, P̃) is considered as a model.

Without lost of generality we will suppose an arrangement of the roots {α_{i}(u), β_{j}(u)} with u ∈ a ⊂ h, such that

0 = α_{1}(u) < α_{2}(u) ≤ α_{3}(u) ≤ … ≤ α_{l_{2}}(u), (3.1)

0 = β_{1}(u) < β_{2}(u) ≤ β_{3}(u) ≤ … ≤ β_{l_{1}}(u). (3.2)

Theorem 2.1. Let P and M be as in the preceding section and let P be a totally geodesic submanifold of M. Suppose that given a unitary vector u ∈ N_{p}P, the following conditions are satisfied for each t ∈ [0, t_{0}] with t_{0} < c(P),

1. K(γ′_{u}(t), H_{1}^{t}) ≥ 0.

2. K(γ′_{u}(t), V_{1}^{t}) ≥ 0.

3. K(γ′_{u}(t), H_{j}^{t}) ≥ n_{j}β_{j}^{2}(u), j = 3, …, l_{1}.

4. K(γ′_{u}(t), V_{i}^{t}) ≥ m_{i}α_{i}^{2}(u), i = 3, …, l_{2}.

5. ∑_{j=2}^{l_{1}} K(γ′_{u}(t), H_{j}^{t}) ≥ ∑_{j=2}^{l_{1}} n_{j}β_{j}^{2}(u).

6. ∑_{i=2}^{l_{2}} K(γ′_{u}(t), V_{i}^{t}) ≥ ∑_{i=2}^{l_{2}} m_{i}α_{i}^{2}(u).

Then, tr S(t) ≥ tr S̃(t) for all t ∈ [0, t_{0}] with t_{0} < inf \{\frac{π}{2β_{j}(u)}, \frac{π}{α_{i}(u)}\}.

Proof. Fix t ∈ [0, t_{0}] and let {E_{j_{k_{1}}}^{⊤} (s), E_{i_{k_{2}}}^{⊥} (s)} defined, for s ∈ [0, t], as in the preceding section. Let {Y_{j_{k_{1}}}^{⊤} (s), Y_{i_{k_{2}}}^{⊥} (s)} be the P -Jacobi fields along γ_{u}(s) satisfying Y_{j_{k_{1}}}^{⊤} (t) = E_{j_{k_{1}}}^{⊤} (t) and Y_{i_{k_{2}}}^{⊥} (t) = E_{i_{k_{2}}}^{⊥} (t). We define vector fields along γ_{u}(s)|_{[0,t]} by

Z_{1_{k_{2}}}^{⊥} (s) = \frac{s}{t}E_{1_{k_{2}}}^{⊥} (s), k_{2} = 2, …, r.
Z_{1_{k_{1}}}^{⊤} (s) = E_{1_{k_{1}}}^{⊤} (s), k_{1} = 2, …, q − r.
Z_{i_{k_{2}}}^{⊥} (s) = s_{α_{i}}E_{i_{k_{2}}}^{⊥} (s), k_{2} = 1, …, m_{i}, i = 2, …, l_{2}.
Z_{j_{k_{1}}}^{⊤} (s) = c_{β_{j}}E_{j_{k_{1}}}^{⊤} (s), k_{1} = 1, …, n_{j}, j = 2, …, l_{1}.

where

(3.3)  s_{α_{i}} = \frac{sin(sα_{i}(u))}{sin(tα_{i}(u))},  c_{β_{j}} = \frac{cos(sβ_{j}(u))}{cos(tβ_{j}(u))}.

From the Index Lemma for submanifolds ([1, page 228]) we have 58 X. Gual-Arnau and R. Masó

\begin{matrix}I_{0}^{t}(Y_{j_{k_{1}}}^{⊤}) ≤ I_{0}^{t}(Z_{j_{k_{1}}}^{⊤}), \\ I_{0}^{t}(Y_{i_{k_{2}}}^{⊥}) ≤ I_{0}^{t}(Z_{i_{k_{2}}}^{⊥}), \end{matrix} (3.4)

where I_{0}^{t} denotes the index form. Moreover,

\begin{aligned}tr S(t) & = & − ∑_{1_{k_{2}}}^{r} I_{0}^{t}(Y_{1_{k_{2}}}^{⊥}) − ∑_{1_{k_{1}}}^{q−r} I_{0}^{t}(Y_{1_{k_{1}}}^{⊤}) \\ & & − ∑_{i=2}^{l_{2}} ∑_{k_{2}=1}^{m_{i}} I_{0}^{t}(Y_{i_{k_{2}}}^{⊥}) − ∑_{j=2}^{l_{1}} ∑_{k_{1}=1}^{n_{j}} I_{0}^{t}(Y_{j_{k_{1}}}^{⊤}). \end{aligned} (3.5)

Therefore, from (3.4) and (3.5),

tr S(t) ≥ − ∑_{1_{k_{2}}}^{r} I_{0}^{t}(Z_{1_{k_{2}}}^{⊥}) − ∑_{1_{k_{1}}}^{q−r} I_{0}^{t}(Z_{1_{k_{1}}}^{⊤})

− ∑_{i=2}^{l_{2}} ∑_{k_{2}=1}^{m_{i}} I_{0}^{t}(Z_{i_{k_{2}}}^{⊥}) − ∑_{j=2}^{l_{1}} ∑_{k_{1}=1}^{n_{j}} I_{0}^{t}(Z_{j_{k_{1}}}^{⊤}).

Since P is totally geodesic,

∑_{1_{k_{1}}}^{q−r} < Z_{1_{k_{1}}}^{⊤} , L_{u}Z_{1_{k_{1}}}^{⊤} > (0) + ∑_{j=2}^{l_{1}} ∑_{k_{1}=1}^{n_{j}} < Z_{j_{k_{1}}}^{⊤} , L_{u}Z_{j_{k_{1}}}^{⊤} > (0) = 0, (3.6)

where L_{u} denotes the Weingarten map of P along the normal vector u; in fact, the Theorem is also true if each of the sums in (3.6) is zero. Therefore,

tr S(t) ≥ − ∫_{0}^{t} \{(r − 1)\frac{1}{t^{2}} + ∑_{i=2}^{l_{2}} m_{i}s′_{α_{i}}^{2} + ∑_{j=2}^{l_{1}} n_{j}c′_{β_{j}}^{2} \} ds

+ ∫_{0}^{t} \{\frac{s}{t}K(γ′_{u}, V_{1}^{t}) + K(γ′_{u}, H_{1}^{t})\} ds

+ ∫_{0}^{t} {∑_{i=2}^{l_{2}} s_{α_{i}}^{2}K(γ′_{u}, V_{i}^{t}) + ∑_{j=2}^{l_{1}} c_{β_{j}}^{2}K(γ′_{u}, H_{j}^{t})} ds

= − ∫_{0}^{t} \{(r − 1)\frac{1}{t^{2}} + ∑_{i=2}^{l_{2}} m_{i}s′_{α_{i}}^{2} + ∑_{j=2}^{l_{1}} n_{j}c′_{β_{j}}^{2} \} ds

+ ∫_{0}^{t} \{\frac{s}{t}K(γ′_{u}, V_{1}^{t}) + K(γ′_{u}, H_{1}^{t})\} ds

+ ∫_{0}^{t} {s_{α_{2}}^{2} ∑_{i=2}^{l_{2}} K(γ′_{u}, V_{i}^{t}) + c_{β_{2}}^{2} ∑_{j=2}^{l_{1}} K(γ′_{u}, H_{j}^{t})} ds

+ ∫_{0}^{t} {∑_{i=3}^{l_{2}}(s_{α_{i}}^{2} − s_{α_{2}}^{2})K(γ′_{u}, V_{i}^{t}) + ∑_{j=3}^{l_{1}}(c_{β_{j}}^{2} − c_{β_{2}}^{2})K(γ′_{u}, H_{j}^{t})} ds.

Finally, from conditions 1.-6. and having in mind that s_{α_{i}}^{2} ≥ s_{α_{2}}^{2} and c_{β_{j}}^{2} ≥ c_{β_{2}}^{2} for

0 < s ≤ t < inf \{\frac{π}{2β_{j}} , \frac{π}{α_{i}} \}, we have

tr S(t) ≥ − ∫_{0}^{t} \{(r − 1)\frac{1}{t^{2}} + ∑_{i=2}^{l_{2}} m_{i}s′_{α_{i}}^{2} + ∑_{j=2}^{l_{1}} n_{j}c′_{β_{j}}^{2} \} ds

+ ∫_{0}^{t} {s_{α_{2}}^{2} ∑_{i=2}^{l_{2}} m_{i}α_{i}^{2} + c_{β_{2}}^{2} ∑_{j=2}^{l_{1}} n_{j}β_{j}^{2}} ds

+ ∫_{0}^{t} {∑_{i=3}^{l_{2}}(s_{α_{i}}^{2} − s_{α_{2}}^{2})m_{i}α_{i}^{2} + ∑_{j=3}^{l_{1}}(c_{β_{j}}^{2} − c_{β_{2}}^{2})n_{j}β_{j}^{2}} ds = tr S̃(t).

□

Corollary 2.1. Under the hypotheses of Theorem 2.1, replacing conditions 3.-4. by one of the following group of conditions: Group 1.

3. K(γ′_{u}(t), H_{j}^{t}) ≤ n_{j}β_{j}^{2}(u), j = 2, …, l_{1} − 1,

4. K(γ′_{u}(t), V_{i}^{t}) ≥ m_{i}α_{i}^{2}(u), i = 3, …, l_{2},

Group 2.

3. K(γ′_{u}(t), H_{j}^{t}) ≥ n_{j}β_{j}^{2}(u), j = 3, …, l_{1},

4. K(γ′_{u}(t), V_{i}^{t}) ≤ m_{i}α_{i}^{2}(u), i = 2, …, l_{2} − 1,

Group 3.

3. K(γ′_{u}(t), H_{j}^{t}) ≤ n_{j}β_{j}^{2}(u), j = 2, …, l_{1} − 1,

4. K(γ′_{u}(t), V_{i}^{t}) ≤ m_{i}α_{i}^{2}(u), i = 2, …, l_{2} − 1,

we obtain the same comparison result for tr S(t). Theorem 2.2. Let P and M be as in Theorem 2.1. Suppose that given a unitary vector u ∈ N_{p}P, the following conditions are satisfied for each t ∈ [0, t_{0}] with t_{0} < c(P),

1. K(γ′_{u}(t), H_{1}^{t}) ≥ 0.

2. K(γ′_{u}(t), V_{1}^{t}) ≥ 0.

3. K(γ′_{u}(t), H_{j}^{t}) ≥ n_{j}β_{j}^{2}(u), j = 2, …, l_{1}.

4. K(γ′_{u}(t), V_{i}^{t}) ≥ m_{i}α_{i}^{2}(u), i = 3, …, l_{2}.

5. ∑_{j=2}^{l_{1}} K(γ′_{u}(t), H_{j}^{t}) + ∑_{i=2}^{l_{2}} K(γ′_{u}(t), V_{i}^{t}) ≥ ∑_{j=2}^{l_{1}} n_{j}β_{j}^{2}(u) + ∑_{i=2}^{l_{2}} m_{i}α_{i}^{2}(u).

Then, tr S(t) ≥ tr S̃(t) forall t ∈ [0, t_{0}] with t_{0} < inf \{\frac{π}{2β_{j}(u)}, \frac{π}{α_{i}(u)}\}.

Proof. As in Theorem 2.1, having into account that c_{β_{j}}^{2} ≥ s_{α_{2}}^{2} and, from (3.1),

s_{α_{i}}^{2} ≥ s_{α_{2}}^{2} for 0 < s ≤ t < inf \{\frac{π}{2β_{j}(u)}, \frac{π}{α_{i}(u)}\}.

□

Corollary 2.2. Under the hypotheses of Theorem 2.2, replacing conditions 3.-4. by one of the following group of conditions: Group 1.

3. K(γ′_{u}(t), H_{j}^{t}) ≥ n_{j}β_{j}^{2}(u), j = 3, …, l_{1}.

4. K(γ′_{u}(t), V_{i}^{t}) ≤ m_{i}α_{i}^{2}(u), i = 2, …, l_{2}.

Group 2.

3. K(γ′_{u}(t), H_{j}^{t}) ≥ n_{j}β_{j}^{2}(u), j = 2, …, l_{1}.

4. K(γ′_{u}(t), V_{i}^{t}) ≤ m_{i}α_{i}^{2}(u), i = 2, …, l_{2} − 1.

Group 3.

3. K(γ′_{u}(t), H_{j}^{t}) ≤ n_{j}β_{j}^{2}(u), j = 2, …, l_{1} − 1.

4. K(γ′_{u}(t), V_{i}^{t}) ≤ m_{i}α_{i}^{2}(u), i = 2, …, l_{2}.

we obtain the same comparison result for tr S(t).

Remark 2.1. When the pair (M̃, P̃) is (CP^{n}(λ), CP^{s}(λ)), Theorem 2.2 and the group of conditions 1 and 2 of Corollary 2.2 give the cases b), a) and c) of Theorem 2.1 in [12], respectively.

Finally, from Theorem 2.1 or Theorem 2.2 we obtain the following result. Corollary 2.3. Let P and M be as in Theorem 2.1. Suppose that given a unitary vector u ∈ N_{p}P, the following conditions are satisfied for each t ∈ [0, t_{0}] with t_{0} < c(P),

1. K(γ′_{u}(t), H_{1}^{t}) ≥ 0.

2. K(γ′_{u}(t), V_{1}^{t}) ≥ 0.

3. K(γ′_{u}(t), H_{j}^{t}) ≥ n_{j}β_{j}^{2}(u), j = 2, …, l_{1}.

4. K(γ′_{u}(t), V_{i}^{t}) ≥ m_{i}α_{i}^{2}(u), i = 2, …, l_{2}.

Then, tr S(t) ≥ tr S̃(t) forall t ∈ [0, t_{0}] with t_{0} < inf \{\frac{π}{2β_{j}(u)}, \frac{π}{α_{i}(u)}\}.

4 Application: Relative volume

Let (M, P) and (M̃, P̃) be as in the preceding sections and suppose that the first conjugate locus of P̃ and the cut-focal locus of P̃ agree (see examples in [2]). Then, if θ_{u}(t) and θ̃_{u}(t) denote the infinitesimal change of volume functions of M and M̃, respectively, from the hypotheses of Theorems 2.1 and 2.2 and Eq. (2.4), having in mind that θ_{u}(0) = θ̃_{u}(0) = 1, we have θ_{u}(t) ≤ θ̃_{u}(t) and, from (2.6) and (2.7), we have c(u) ≤ z(θ_{u}(t)) ≤ z(θ̃_{u}(t)) = c̃(u), (4.1)

and therefore,

Vol(M) = ∫_{P} ∫_{S^{n−s−1}(1)} ∫_{0}^{c(u)} t^{n−s−1}θ_{u}(t)dt du dP

≤ ∫_{P} ∫_{S^{n−s−1}(1)} ∫_{0}^{_{c}(u)} t^{n−s−1}θ̃_{u}(t)dt du dP.

≤ (Vol(P)/Vol(P̃)) ∫_{P} ∫_{S^{n−s−1}(1)} ∫_{0}^{_{c}(u)} t^{n−s−1}θ̃_{u}(t)dt du dP̃
= (Vol(P)/Vol(P̃))Vol(M̃).

(4.2)

So, we conclude the following inequality between the relative volumes:

Theorem 3.1. Let (M, P) and (M̃, P̃) as before; then,

(4.3) \frac{Vol(P̃)}{Vol(M̃)} ≤ \frac{Vol(P)}{Vol(M)}.

Appendix. Suppose now that M̃ is a noncompact symmetric space; then, having into account the duality between compact and noncompact symmetric spaces, the main differences with the compact case are that the Ricci curvatures defined in (2.11) and (2.12) are negative and all the trigonometric functions which appear for the compact case have to be changed to hyperbolic functions. Therefore, the comparison results in Section 2 remain valid for noncompact symmetric spaces but some of the inequalities imposed in the different conditions change. For instance, in Theorem 2.1, Corollary 2.1 and Corollary 2.3, having in mind that s_{α_{i}}^{2} ≤ s_{α_{2}}^{2} and c_{β_{j}}^{2} ≤ c_{β_{2}}^{2} for

0 < s ≤ t < inf \{\frac{π}{2β_{j}} , \frac{π}{α_{i}} \}, when hyperbolic functions are considered, we have to

change all the inequalities ≤ by ≥ and vice versa, to obtain the comparison results for tr S(t). But, in Theorem 2.2 and Corollary 2.2, some of the inequalities change

but other remain valid because c_{β_{j}}^{2} ≤ s_{α_{i}}^{2} when 0 < s ≤ t < inf \{\frac{π}{2β_{j}} , \frac{π}{α_{i}} \}, also for

hyperbolic functions.

However, concerning the application in Section 4, when M̃ is noncompact, the totally geodesic submanifold P̃ of M̃ is also a noncompact symmetric space, therefore, Theorem 3.1 has no sense because we can not obtain finite values for the different volumes, [8]. Acknowledgements.Work partially supported by a DGES Grant BSA2001-0803-C02-02.

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Departament de Matemàtiques, Campus Riu Sec, Universitat Jaume I, 12071-Castelló. Spain email: gual@mat.uji.es