Equality sign holds if and only if x is either of 1-type and of order [1, 1] or of order [2, 2] or x is of ..... STATE UNIVERSITY. EAST LANSING, MICHIGAN 48824.
B.-Y. CHEN KODAI MATH. J. 10 (1987), 93-101
SOME ESTIMATES OF TOTAL TENSION AND THEIR APPLICATIONS BY
BANG-YEN CHEN
Abstract In this note, we give two best possible estimates of the total tension for a smooth map. Such estimates are established in terms of order of the map. Applications of such estimates to isometric immersions and to spectral geometry are given by applying an inequality obtained in [3]. 1. Introduction. Let M be a compact submanifold of a Euclidean ra-space Em. By applying the induced metric on M, the author introduced in [2] the notion of order of the submanifold. The notion of order is known to be closely related with the differential geometry of the submanifold (cf. [4]). In [5, 6] such notion was generalized to smooth maps of a compact Riemannian manifold into Em. Some relations between the total tension and the order were obtained in [5, 6]. In this note, we will obtain two more relations between the total tension and the order of a map. Such relations are applied to obtain a best possible estimate of the total mean curvature of a spherical submanifold. By using a best possible inequality derived in [3], such relations were then applied to obtain some best possible eigenvalue estimates for minimal submanifolds in rankone symmetric spaces. 2. Order of a Map. Let M be a compact n-dimensional Riemannian manifold and Δ the Laplacian of M acting on the space C°°(M) of smooth functions. Then Δ has an infinite discrete sequence of eigenvalues: 0 = ^ o < ^ < ^ 2 < ••• £ , we can apply the above decomposition to m each coordinate function of M in E . Thus, we obtain the following spectral m decomposition of the £ -valued function x : (O Λ\
r
__rι_Vr ί=l
where ;c0 is a constant vector which is the center of mass of x and xt a vector with Axt~λtxt. If x is a non-constant map, then there exists a positive integer p such that xpφ0 and %=JC O +Σί^p^ί. If there are infinitely many nonzero xt's in the decomposition (2.1), we put q—co. Otherwise, we let q be the largest integer such that xqφ0 in the decomposition (2.1). In both cases we have /O O\
-
I
γ
"VΛ
γ
We call \_p, q~\ the order of the map x. The map is said to be of finite type if q is finite. Otherwise, the map is said to be of infinite type. More precisely, the map x is said to be of k-type (k2), SO(10)/£/(5), and £ 6 /Spin(10)xT, with an appropriate metric, and m is given by k, k{k+3)/2, k+1, k{k+2), k(k+3)/2, 15, and 26, respectively. Proof. Let z=\z0> •••, zm)^Fm+1. We denote by H(m+1;F) the space of all ( m + l ) x ( m + l ) Hermitian matrices over F. On H(m+1;F) we define an inner product < , > by =(1/2) Re tr (ΛB). On Fm+1 we consider the metric =Re (^z'), *( ) denotes the transpose. Let S ( m + 1 ) c ί - 1 denote the unit hypersphere of Fm+1 defined by {z—l}. Then the projective mspace FPm can be regarded as the quotient space of the unit hypersphere obtained by identifying t(z0, •••, zm) with t(cz0, •••, czm) with c^F and | c | = l . Define a mapping p: S ( m + 1 ) ί * - W / ( r a + l ; F) by (4.10)
p{z)=zz*, ι
where z^— z. If F=Ry p defines an isometric immersion of Sm into H(m+1 R) m and it induces an isometric imbedding p of RP into H(m+1; R). If F—C or m jfiΓ, p induces an isometric imbedding p of FP into H(m+1; F). m If M admits a minimal isometric immersion into FP , then by regarding m FP as a submanifold of H(m+1; F) via p, we have the following best possible inequality of the mean curvature of M in H(m+1 F) (Lemma 2 of [3] or Lemma 6.5 of [4, p. 152]): (4.11)
\H\
where H denotes the mean curvature vector of M in H(m+1;F). Moreover, from Lemma 2 of [3], we know that equality sign of (4.11) holds if and only if M is a quaternion submanifold if F—H\ M is a complex submanifold if F=C; and if F=R, then the equality sign of (4.11) holds automatically. On
SOME ESTIMATES OF TOTAL TENSION
the other hand, it is known that FPm is imbedded in a hypersphere of H(m+1 with radius r=Vm/2(m+l) via p ([12] or cf. [3, 4]). rem 3 and inequality (4.11), we may obtain (4.12)
99
F)
Thus, by applying Theo-
™
This implies (4.9). If F~H and the equality sign of (4.9) holds, then the equality sign of (4.11) holds. Hence, Lemma 2 of [3] shows that M is a quaternionic submanifold of HPm. Because the only quaternionic submanifolds of HPm are quaternionic totally geodesic submanifolds. Thus, we conclude that M is a HPnlA. Now, since λ1 and λ2 of HPk are given by S(k+ΐ) and 8(2&+3), respectively. Thus, we obtain n=4m. The converse of this is easy to verify. If F—C and the equality sign of (4.9) holds, then the equality sign of (4.11) holds. Thus, Lemma 2 of [3] implies that M is a Kaehler submanifold of CPm. Moreover, from Theorem 3, we see that either M is of 1-type in H(m+1; C) or M is of 2-tyρe and of order [1, 2]. If M is of 1-type, then by a result of Ros [10], we know that M i s a totally geodesic CPk (2k —n). Since λx and λ2 of CPk are given by 4(&+l) and 8(&+2), respectively, we find n—2m. If M i s of 2-type and of order [1, 2], then we may apply a result of Ros-Udagawa [10, 13] about the classification of compact Kaehler submanifolds of CPm of order [1, 2], Such submanifolds are exactly non-totally geodesic Kaehler submanifolds which are Einsteinian and with parallel second fundamental form (cf. Proposition 3 of [13]). Furthermore, such Kaehler submanifolds were classified by Nakagawa and Takagi [ 9 ] ; they are CPk(2), Q\ CP*(4)xCP*(4), U(k+2)/ U(k)xU(2) (k>2), SO(10)/U(5) and £ 6 /Spin (10)xT which lie fully in CPm with m given respectively by k(k+3)/2, k+1, k(k+2), k(k+3)/2, 15 and 26, respectively. Conversely, if M is one of Einstein Hermitian symmetric spaces and m is the corresponding integer, then by the known values of λλ and λ2 of these spaces (see Table 1 below), we see that the equality sign of (4.9) holds. This completes the proof. Remark 2. If F—R and the equality sign of (4.9) holds, then M is of order [1, 2] in H(m+1; R) by Theorem 3. If M is a projective space FPk or the Cayley plane and if σ: M-^SN is the first standard imbedding of M, then it is clear that the composite immersion p°σ : M-^SN-^H(nJΓl R) is of order [1, 2]. Moreover, if σ is full, then the equality sign of (4.9) holds. In view of Theorem 4 and [1], it seems to be interesting to classify all compact minimal subm manifolds of RP which satisfy the equality sign of (4.9). From Theorem 4 we also have the following. COROLLARY 1. If M is a compact n-dimensional minimal submanif old of m 5 (l), then we have
100
BANG-YEN CHEN
(4.13)
λ1λ2^2n(m+l){λ1+λ2-2n-2}.
m
m
Proof. If M admits a minimal isometric immersion into S (l), then it m admits a minimal isometric immersion into RP . Thus, (4.13) follows immediately from Theorem 4. Remark 3. Ros [11] obtain a best possible inequality between λx and λ2 similar to (4.13) with an additional assumption that M admits an order 1 minimal immersion in a sphere (see, also [8]). Remark 4. In [10, 13], the Einstein Hermitian symmetric spaces given in Table 1 were characterized by their spectrum among all compact Kaehler subm manifolds of CP . By applying Theorem 4, we see that these manifolds can be characterized by their spectrum among all compact minimal submanifolds of CPm. Table 1. Submanifold
n
m
CP*(4)
2k
k
CP\2)
2k
Qh CP*(4)xCP*(4)
2k 4k
U(k+2)/U(k)xU(2) k>2 SO(10)/C/(5) £ 6 /Sρin(10)xT
4(6+1)
8(6+2)
2(6+1)
4(6+2)
k+l 4(6+2)
46 4(6+1)
4(6+2) 8(6+1)
46
y6(6+3)
4(6+2)
8(6+1)
20 32
15
32
26
48
jk(k+3)
48 72
REFERENCES [ 1 ] M. BARROS AND B. Y. CHEN, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, to appear in Nagoya Math. J. 108 (1987). [ 2 ] B. Y. CHEN, On the total curvature of immersed manifolds, IV, Bull. Math. Acad. Sinica, 7 (1979), 301-311; , VI, ibid., 11 (1983), 309-328. [ 3 ] B. Y. CHEN, On the first eigenvalue of Laplacian of compact minimal submanifolds of rank one symmetric spaces, Chinese J. Math., 11 (1983), 259-273. [ 4 ] B. Y. CHEN, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, 1984. [ 5]
B. Y. CHEN, J. M. MORVAN AND T. NORE,
Energie, tension, et ordre des applica-
tions a valeur dans un espace euclidien, C. R. Acad. Sc. Paris, 301 (1985), 123126. [ 6 ] B. Y. CHEN, J. M. MORVAN AND T. NORE, Energy, tension and finite type maps,
SOME ESTIMATES OF TOTAL TENSION [7] [ 8] [ 9] [10] [11] [12] [13]
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Kodai Math. J. 9 (1986), 406-418. J. EELLS AND L. LEMAIRE, A report on harmonic maps, Bull. London Math. Soc, 10 (1978), 1-68. S. MONTIEL, A. Ros AND F. URBANO, Curvature pinching and eigenvalue rigidity for minimal submanifolds, Math. Z., 191 (1986), 537-548. H. NAKAGAWA AND R. TAKAGI, On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Soc. Japan, 28 (1976), 638-667. A. Ros, On spectral geometry of Kaehler submanifolds, J. Math. Soc. Japan, 36 (1984), 433-448. A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z., 187 (1984), 393-404. K. SAKAMOTO, Planar geodesic immersions, Tohoku Math. J., 29 (1977), 25-56. S. UDAGAWA, Spectral geometry of Kaehler submanifolds of a complex projective space, J. Math. Soc. Japan 38 (1986), 453-472. DEPARTMENT OF MATHEMATICS MICHIGAN STATE UNIVERSITY EAST LANSING, MICHIGAN 48824
USA
