Riemannian DNA, Inequalities and Their Applications

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1 Tamkang Journal of Science and Engineering, Vol. 3, No. 3, pp (2000) 123 Riemannian DNA, Inequalities and Their Applications Bang-Yen Chen Department of Mathematics, Michigan State University, East Lansing, Michigan , USA Abstract The main purpose of this survey article is to present the new type of Riemannian curvature invariants (Riemannian DNA) and the sharp inequalities, involving these invariants and the squared mean curvature, originally introduced and established in [7,8]. These Riemannian DNA affect the behavior in general of the Riemannian manifold and they have several interesting connections to several areas of mathematics. For instance, they give rise to new obstructions to minimal and Lagrangian isometric immersions. Moreover, these invariants relate closely to the first nonzero eigenvalue of the Laplacian on a Riemannian manifold. These invariants together with the sharp inequalities gives rise naturally to the notion of ideal immersions or the notion of the best ways of living. We also explain the physical meaning of the notion of ideal immersions for Riemannian manifolds in a Riemannian space form based again on the sharp inequalities. Key Words: Riemannian invariants, squared mean curvature, tension, ideal immersion, best way of living, Riemannian DNA 1. Introduction The theory of submanifolds have been studied since the invention of calculus and it was started with curvature of plane curves. For a surface in Euclidean 3-space one has the two important quantities, namely, the mean curvature and the Gauss curvature. The mean curvature is an extrinsic invariant which measures the surface tension of the surface arisen from the ambient space The theorema of C. F. Gauss which asserts the invariance of the Gauss curvature under isometric deformations of surfaces which live in the Euclidean world is egregious as C. F. Gauss labeled it in his general theory of curved surfaces (1827). Its impact on the development of mathematics has been equally egregious indeed. Immediately, this theorem lead to the distinction between the intrinsic and the extrinsic qualities of such surfaces. Later, the awareness of the existence of an intrinsic geometry of surfaces in Euclidean 3-space resulted in the creation of Riemannian manifolds; which later provides the mathematical foundation of Einstein's relativity theory. The Riemannian geometry forms the theory of modern differential geometry, as expressed by S. S. Chern in his 1970 talk at the international congress of mathematics at Nice. Riemannian invariants are the intrinsic characteristics of the Riemannian manifolds. Riemannian invariants of a Riemannian manifold affect the behavior in general of the Riemannian manifold. Borrow a term from biology, Riemannian invariants are the DNA of Riemannian manifolds. In the words of M. Berger [1]: Curvature invariants are N o 1 Riemannian invariants and the most natural. Gauss and then Riemann saw it instantly. Curvature invariants also play key roles in physics. For examples, the magnitude of a force required to move an object at constant speed, according to Newton's laws, a constant multiple of the curvature of the trajectory. The motion of a

2 124 B. Y. Chen body in a gravitational field is determined, according to Einstein, by the curvature of the space time. All sorts of shapes from soap bubbles to red blood cells, seems to be determined by various curvatures [22]. Besides sectional curvatures, the Ricci curvature and the scalar curvature have been the most studied scalar valued curvature notions on Riemannian manifolds. Once Riemannian spaces were around, the differential geometry of surfaces in Euclidean 3- space was generalized to the differential geometry of higher dimensional submanifolds of Riemannian manifolds. In this general theory of Riemannian submanifolds, the following problem is fundamental. Problem 1. Establish simple relationship between the main intrinsic invariants and the main extrinsic invariants of the submanifolds. The abstract mathematical theory of Riemannian manifolds amply proved their relevance for understanding better many diverse types of experiences related to real world situations which are investigated in many areas of science. In particular, the famous isometric embedding theorem of J. F. Nash (1954) shows the realisability as submanifolds of Riemannian manifolds in Euclidean spaces. Nash s theorem was aimed for, in particular, in the hope that it would made possible to derive new intrinsic results, taking profit from the fact that if so that Riemannian manifolds could always be considered as submanifolds of Euclidean spaces, this would then yield the opportunity to use extrinsic help. Another observation concerning Nash s embedding theorem was made by S. T. Yau (Mathematical research today and tomorrow, Viewpoints of seven fields medalists, Springer- Verlag 1991, pp ) as follows: Higher-dimensional submanifolds are much more difficult to understand. Despite the famous Nash isometric embedding theorem, we still do not know how to isometrically embed an abstract manifold in a nice manner. What is lacking in the Nash theorem is the control of the extrinsic quantities in relation to the intrinsic quantities. The rigidity is also far from being understood. In this line of thought, S. S. Chern formulated already in 1968 the following specific question [15]: Search for new intrinsic obstructions to the existence of minimal immersions of a Riemannian manifold into a Euclidean space, besides positivity of Ricci curvature. 2. New type of Riemannian invariants Let M be a Riemannian n-manifold. Denote by K(V) the sectional curvature of M associated with a plane section V of the tangent space of M at a point p in M. For an orthonormal basis e 1,...,e n of the tangent space, the scalar curvature τ at p is defined to be τ = K(e i Λe j ). (2.1) i < j Let L be a subspace of T p M of dimension r < n. We denote by τ(l) the scalar curvature of L. The scalar curvature τ(m) of M at p is nothing but the scalar curvature of the tangent space of M at p. And if L is a 2-plane section, τ(m) is nothing but the sectional curvature of L. For an integer k 0, denote by S(n,k) the finite set consisting of k-tuples (n 1 ) of integers 2 satisfying n 1 2 and n n k n. Denote by S(n) the set of k- tuples with k 0 for a fixed n. The cardinal number #S(n) of S(n) increases quite rapidly with n. For instance, for n = 2, 3, 4, 5, 6, 7, 8, 9, 10,..., 20,, 50,, 100, the cardinal number #S(n) is given respectively by 1, 2, 4, 6, 10, 14, 21, 29, 41,, 626,, ,..., For each k-tuple (n 1 ) we define an invariant δ(n 1 ) by δ(n 1 )(p) = τ (p) inf(τ(l 1 ) τ (L k )) where L 1,..., L k run over all k mutually orthogonal subspaces of T p M such that dim L i = n i,i = 1,...,k. In particular, we have δ( ) = τ, δ(2) = τ inf K.

3 Riemannian DNA, Inequalities and Their Applications 125 The invariants δ(n 1 ) with k > 0 and the scalar curvature τ are very different in natural. 3. Sharp inequalities For each k-tuple (n 1 ) in S(n), we define two positive numbers: c (n 1,..., n k ) = n 2 (n + k 1 n j ), 2(n + k n j ) b(n 1 ) = 1 (n(n 1) 2 n k j j=1 (n j 1)). The following theorem is fundamental. Theorem 3.1. [7,8] Let M be an n-dimensional submanifold in a Riemannian space form of constant curvature c. Then, for each k-tuple (n 1 ) in S(n), we have δ (n 1,..., n k ) c(n 1,..., n k )H 2 + b(n 1,..., n k )c. (3.1) where H 2 is the squared mean curvature. The equality sign of (3.1) holds if and only if the shape operator of M satisfies some special form. 4. Some applications Theorem 3.1 (i) An application to Yau s question: Inequalities (3.1) provide a sharp relationship between intrinsic quantities and the most important extrinsic quantity H 2 for submanifolds with arbitrary codimension. Hence, it provides an answer to Yau s question mentioned in Introduction. (ii) An application to Chern s question: The following theorem provides a solution to Chern s question. Theorem 4.1. Let M be a Riemannian n-manifold. If there exists a k-tuple (n 1 ) in S(n) such that δ(n 1 )(p) > 0 at a point p in M, then M admits no minimal isometric immersion into a Euclidean space for any codimension. (iii) An application to symplectic geometry: For Lagrangian immersions in complex Euclidean n-space C n, a result of M. Gromov [20] states that a compact n-manifold M admits a Lagrangian immersion (not necessary isometric) in C n if and only if the complexification TM C of the tangent bundle of M is trivial. Gromov's result implies that there do not exists any topological obstruction to Lagrangian immersions for compact 3-manifolds in a complex Euclidean 3-space, because the tangent bundle of a 3-manifold is trivial. From Riemannian point of views, it is natural to ask the following basic question: What are the necessary conditions for a compact Riemannian manifold to admit a Lagrangian isometric immersion into a complex Euclidean space? Our invariants do provide many sharp Riemannian obstructions to Lagrangian isometric immersions as follows: Theorem 4.2. Let M be a compact Riemannian n- manifold with finite fundamental group π 1 (M) or with null first betti number b 1 (M). If δ(n 1 ) > 0 for some (n 1 ) in S(n), then M cannot isometrically immersed in the complex Euclidean n-space C n as a Lagrangian submanifold. (iv) New intrinsic result via Nash s theorem: We may apply Theorem 3.1 to prove the following intrinsic result. Theorem 4.3. Let M be an irreducible compact homogeneous Riemannian n-manifold. Then the first nonzero eigenvalue λ 1 of the Laplacian on M satisfies λ 1 n (n 1 ) (4.1) for any k-tuple (n 1 ) in S(n), where (n 1 ) = δ(n 1 ) c(n 1 ). (4.2) Remark 4.1. Inequality (4.1) extends a wellknown result of T. Nagano [21] who proved (4.1)

4 126 B. Y. Chen for k = 0, i.e., λ 1 nρ, with the equality holding if and only if M is a Riemannian n-sphere. Where ρ is the normalized scalar curvature. (v) A new viewpoint of rigidity : Theorem 4.1 can also be applied to obtain many rigidity theorems for submanifolds with arbitrary codimension without any global assumption. Our argument woks as follows: For a submanifold M in any Euclidean space, the inequality H 2 (n 1,..., n k ) provides a lower bound of the squared mean curvature. When the inequality is actually an equality, Theorem 3.1 guarantees the special form of the shape operators for the submanifold. In many cases, this information on the Riemannian structure of M and on the shape operators are sufficient to obtain the rigidity of the submanifold. Here we just mention two of many such applications. Theorem 4.4. Let M be an open portion of a unit n-sphere. Then for any isometric immersion of M into a Euclidean space with any codimension, we have H 2 1. (4.3) The equality case holds identically if and only if M is immersed as an open portion of an ordinary hypersphere in an affine (n+1)-subspace. Theorem 4.5. Let M be an open portion of S k (1) E n k, k > 1. Then, for any isometric immersion of M into a Euclidean m-space with arbitrary codimension, we have H 2 (k / n) 2. (4.4) The equality case holds identically if and only if M is immersed as an open portion of an ordinary spherical hypercylinder in an affine (n+1)-subspace of E m. 5. What is an ideal immersion? Definition 5.1. An isometric immersion x : M n R m (c) from a Riemannian n-manifold M n into a Riemannian space form of constant sectional curvature c is called an ideal immersion if the equality case of (3.1) holds identically, i.e., δ(n 1 ) = c(n 1 )H 2 + b(n 1 )c (5.1) for some k-tuple (n 1 ) in S(n). Remark 5.1 (Physical interpretation of ideal immersions) An isometric immersion of M into a Riemannian space form is an ideal immersion means that M lives in Riemannian space form in the most comfortable way, in the sense that M receives the least possible amount of tension from the surrounding space at each point p on M. This is due to (3.1), (5.1) and the fact that the mean curvature vector field is exactly the tension field for an isometric immersion (a well-known fact since the time of Laplace). Thus, the squared mean curvature at each point on the submanifold simply measures the amount of tension the submanifold receiving from the surrounding space at that point. If one defines a best world as a complete Riemannian space with the highest degree of homogeneity then, according to the work of Lie, Klein and Killing, the family of best worlds consists of Euclidean spaces, spheres, real projective spaces, and real hyperbolic spaces. These spaces have the highest degree of homogeneity, since their groups of iisometries have the maximal possible dimension. In this sense. a best world is nothing but a Riemannian space form. For the above reasons, an ideal immersion is also called a best way of living in a best world. Remark 5.2. (Ideal immersions are stable critical points of the total squared mean curvature functional) Ideal immersions are closely related to critical point problem in the study of total mean curvature. This can be seen as follows: Let M be a compact Riemannian manifold (with or without boundary). Denote by I(M) the family of isometric immersions of M into a Riemannian space form. If f is an isometric immersion of M into a Riemannian space form, we define its total mean curvature (or total tension) by the formula [4]: TMC( f ) = H n dv, (5.2) M

5 Riemannian DNA, Inequalities and Their Applications 127 where H 2 denotes the squared mean curvature of f, then it follows from Theorem 3.1 that an ideal immersion of M is a critical point of the functional of total mean curvature within the class of I(M). Moreover, every ideal immersion is stable in the class of I(M). 6. Examples of ideal immersions The class of ideal submanifolds is huge. Here, we provide several very simple examples of ideal submanifolds. Example 6.1. Every totally umbilical submanifold of a real space form is an ideal immersion. In particular, a horosphere in a hyperbolic space is an ideal submanifold. Example 6.2. For an integer k, 0 k [ n ], the 2 spherical cylinder E k S n k is an ideal hypersurface of Euclidean (n+1)-space. Example 6.3. Every homogeneous minimal hypersurface of (n+1)-sphere with three distinct principal curvatures is ideal. Example 6.4. Every austere hypersurface, in particular, every minimal real Kaehler hypersurface of a real space form, is an ideal hypersurface. By a real Kaehler hypersurface we mean a real codimension one isometric immersion of a Kaehler manifold. A submanifold is called austere in the sense of Harvey and Lawson, if the set of eigenvalues of each shape operator is invariant under multiplication by A basic problem Not every Riemannian manifold admits an ideal immersion into some space form, for example, the real projective plane does not admit any ideal immersion in any Euclidean space, although locally it does. Hence, a basic problem in the study of ideal immersions is the following. Problem 7.1. Determine ideal immersions of all Riemannian manifolds in a space form. In other words, to solve the following. World Problem. Determine the best ways of living for all individual Riemannian manifolds who live in a best world. This is in fact a very difficult problem in general. So far, this problem was solved only for some special families of Riemannian manifolds which we will present in the next section. 8. Some solutions to Problem 7.1 Problem 7.1 has been solved for the following special families of Riemannian submanifolds. (a) The families of all Riemannian 2-manifold. This family was done by Chen in [5]. (b) The families of real space forms. This family is done by Chen, Dillen, Verstraelen and Vrancken by the following. Theorem 8.1 [12] An isometric immersion from a Riemannian space form into another Riemannian space form is ideal if and only if it is totally umbilical. (c) The families of compact irreducible homogeneous Riemannian spaces. This family is done by Chen by the following theorem. Theorem 8.2. [8] A compact irreducible homogeneous Riemannian manifold M admits an ideal immersion in a Euclidean space if and only if λ 1 = nω, (8.1) where λ 1 is the first positive eigenvalue of the Laplacian and Ω is defined by Ω = max (n 1 ), (n 1 ) being taken over all k-tuples in S(n). (d) The family of Lagrangian submanifolds in complex space forms.

6 128 B. Y. Chen This family was classified by Chen in [9]. (e) The family of Lagrangian submanifolds in the nearly Kaehler 6-sphere [19]. Dillen and Vrancken proved that such ideal submanifolds are tubes of radius π / 2 in the direction of the second normal space over an almost complex curves in the unit 6-sphere. (f) The family of conformally flat ideal hypersurfaces [11,14]. In particular, it was proved by Chen, Dillen and Verstraelen that that there exist 31 classes of conformally flat manifolds which can be isometrically immersed in some Riemannian space form as ideal hypersurfaces (there exist 7 classes as ideal hypersurfaces in Euclidean spaces; 6 classes in spheres; and 18 classes in hyperbolic spaces). Many of such conformally flat ideal hypersurfaces are related with Jacobi's elliptic functions and hypergeometric functions. (g) Some partial solutions for the family of CRsubmanifolds in complex space forms. These solutions are obtained Chen, Vracken in [13] and Sasahara [13,23]. (h) Many other results related with ideal immersions have also been obtained by various authors, e.g., D. E. Blair, J. Bolton, A. Carriazo, B. Y. Chen, M. Dajczer, F. Defever, R. Deszcz, F. Dillen, L. A. Florit, Y. Hong, C. S. Houh, Y.-H. Kim, D.-S. Kim, K. Matsumoto, I. Mihai, M. Petrovic, T. Sasahara, C. Scharlach, L. Verstraelen, L. Vrancken, L. M. Woodward, J. Yang, S. Yaprak, among others. Most results obtained by them are related to (2)-ideal submanifolds in various space forms. In particular, they derived the corresponding inequalities for submanifolds in Sasakian, Kaehlerian or quaternionic space forms. They also investigated some submanifolds in such space forms which verify the equality case of the corresponding inequalities. (i) Recently, M. Kriele, C. Scharlach, U. Simon, L. Verstraelen and L. Vrancken have studied affine hypersurfaces which satisfy an affine version of equality (3.1) assciated with the simplest 1-tuple (2). In fact, they derived the corresponding inequality of (3.1) associated with the simplest 1- tuple, namely, (2), for affine hypersurfaces. Furthermore, they also investiaged the affine hypersurfaces which satisfy the equality case of the corresponding inequality. In other words, they studied the affine version of (2)-ideal hypersurfaces in affine spaces. 9. Some open problems Here, we propose to investigate the following open problems related to the new Riemannian invariants and ideal immersions. Problem 9.1. To discover further solutions to the classification problem of ideal immersions. Problem 9.2. To discover some topological implications of the new Riemannian invariants. Problem 9.3. To discover further applications of the new invariants and of ideal immersions. Problem 9.4. To establish further relationship between these new type of Riemannian DNA and other quantities. For instance, to discover relationship between these invariants with the root system of a symmetric space, index of the critical points,..., etc. For a compact Riemannian n-manifold M and any isometric immersion of M into any Euclidean space, one has [3,4]: dv vol(s n (1)). (9.1) H n M Let i(m) = inf{tmc( f ): f I(M)}. If M admits an ideal immersion j of M into Euclidean space, then we have TMC(j) = i(m). Problem 9.5. To study the invariant i(m) for Riemannian manifolds. In particular, to study the following two problems. Problem 9.6. To find a necessary and sufficient condition for a Riemannian manifold M to admit an isometric immersion x which satisfies TMC(x) = i(m).

7 Riemannian DNA, Inequalities and Their Applications 129 Problem 9.7. To discover further relations between i(m) and the new Riemannian invariants on a Riemannian manifold M. 10. Discussion (i) We shall point out that not every Riemannian DNA of a Riemannian manifold relates to the tension the submanifold receiving from ambient space, although every one of the Riemannian invariants we defined in section 2 does. (ii) In views of (i) it is very interesting to determine whether there exist Riemannian DNA on a Riemannian manifold M, other than those introduced in section 2, which are directly related to the lease amount of tension which M must receive from the ambient space whenever M is isometrically immersed in some Riemannian space form. (iii) Beside inequality (3.1) there is another direct sharp relationship between some intrinsic quantities and extrinsic quantities for an arbitrary submanifold in a Riemannian space form; namely, there exists a direct relationship between the k- Ricci curvature and the shape operator A H of an arbitrary submanifold M in a Riemannian space form with arbitrary codimension given as follows. For each integer k,2 k dimm, let θ k denote the Riemannian invariant defined on M by 1 θ k = ( k 1 ) inf Ric L(X), (10.1) where L runs over all k-plane section in the tangent space T x M, x M, and X runs over al unit vectors in L. Then it was proved in [6] that for any isometric immersion of M into any Riemannian space form R m (c) of constant curvature c, one has A H > n 1 n (θ k c)i, (10.2) where I denotes the identity map of the tangent bundle, whenever θ k c. And A H 0, (10.3) whenever θ k = c. Inequalities (10.2) and (10.3) provide us another solution to Problem 1 mentioned in Introduction. Inequalities (10.2) imply that if there is a k, 2 k n, such thatθ k > c at some point x in M, then, for every isometric immersion of M into any Riemannian space form of constant curvature c, each eigenvalue of A H at x is greater than (n-1)/n. Also (10.3) implies that every hypersurface with positive k-ricci curvature in a Riemannian space form is convex (cf. [6,9] for details). Reference [1] Berger, M., La geometrie metrique des varietes Riemnniennes, Elie Cartan et les Mathematiques d Aujourd Hui, Asterisque, pp (1985). [2] Blair, D. E., Dillen, F., Verstraelen, L. and Vrancken, L., Calabi curves as holomorphic Legendre curves and Chen s inequality," Kyunpook Math. J. Vol. 35, Jun, pp (1995). [3] Chen, B. Y., On the total curvature of immersed manifolds I, Amer. J. Math. Vol. 93, pp (1971). [4] Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publ., Co. River Edge, NJ (1984). [5] Chen, B. Y., Mean curvature and shape operator of isometric immersions in realspace-forms, Glasgow Math. J. Vol. 38, pp (1996). [6] Chen, B. Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. Vol. 41, pp (1999). [7] Chen, B. Y., Strings of Riemannian invariants, inequalities, ideal immersions and their applications, Third Pacific Rim Geom. Conf., Intern. Press, Cambridge, MA, pp (1998). [8] Chen, B. Y., Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. Vol. 26, pp (2000). [9] Chen, B. Y., Ideal Lagrangian immersions in complex space forms, Math. Proc. Cambridge Phil. Soc. Vol. 128, pp (2000).

8 130 B. Y. Chen [10] Chen, B. Y., Riemannian submani-folds, Handbook of Differential Geometry, Vol. I, (North Holland Publ.) pp (2000). [11] Chen, B. Y., Dillen, F. and Verstraelen, L., Conformally flat ideal hypersurfaces, (preprint). [12] Chen, B. Y., Dillen, F. Verstraelen, L. and Vrancken, L., Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces, Proc. Amer. Math. Soc. Vol. 128, pp (2000). [13] Chen, B. Y.and Vrancken, L., CRsubmanifolds of complex hyperbolic spaces satisfying a basic equality, Israel J. Math. Vol. 110, pp (1999). [14] Chen, B. Y. and Yang, J., Elliptic functions, theta function and hypersurfaces satisfying a basic equality, Math. Proc. Cambridge Phil. Soc. Vol. 125, pp (1999). [15] Chern, S. S., Minimal Submanifolds in a Riemannian Manifold, Univ. of Kansas, Lawrence, Kansas (1968). [16] Dajczer, M. and Florit, L. A., On Chen s basic equality, Illinois J. Math. Vol. 42, pp (1998). [17] Defever, F., Mihai, I. and Verstraelen, L., B. Y. Chen s inequality for C-totally real submanifolds in Sasakian space forms, Boll. Un. Mat. Ital. Ser. B Vol. 11, pp (1997). [18] Dillen, F., Petrovic, M., and Verstraelen, L., Einstein, conformally flat and semisymmetric submanifolds satisfying Chen s equality, Israel J. Math. Vol. 100, pp (1997). [19] Dillen, F. and Vrancken, L., Totally real submanifolds in 6-sphere satisfying Chen s equality, Trans. Amer. Math. Soc. Vol. 348, pp (1996). [20] Gromov, M., A topological technique for the construction of solutions of differential equations and inequalities, Intern. Congr. Math. Nice 1970, Vol 2, pp , (1971). [21] Nagano, T., On the minimum eigenvalues of the Laplacians in Riemannian manifolds, Sci. Papers College Gen. Edu. Univ. Tokyo, Vol 11, pp , (1961). [22] Osserman, R., Curvature in the eighties, Amer. Math. Monthly, Vol 97, pp (1990). [23] Sasahara, T., CR-submanifolds in complex hyperbolic spaces satisfying an equality of Chen, Tsukuba J. Math. Vol. 23, pp (1999).