Tadashi Nagano

Bio: Tadashi Nagano is an academic researcher from University of Notre Dame. The author has contributed to research in topics: Geodesic map & Compact Riemann surface. The author has an hindex of 9, co-authored 11 publications receiving 475 citations. Previous affiliations of Tadashi Nagano include University of Bonn.

Totally geodesic submanifolds of symmetric spaces, I

Abstract: One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M introduced and studied by Chen and Nagano in (Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405-425) as minimal submanifolds. The other purpose is to establish a stability theorem for minimal totally real sub- manifolds of Kahlerian manifolds.

Transformation groups on compact symmetric spaces

Abstract: The symmetric spaces constitute the most important class of Riemannian manifolds; some of them have been the standard spaces in various branches of geometry, and many authors threw light upon their deep properties. Still there seems to be no thorough study of general transformation groups L (other than the isometry groups H, but containing G) of compact(2) symmetric spaces M, which we call geometric transformation groups of M in this introduction. Its need will be patent if it will reveal interrelations of symmetric spaces, and if L will be the automorphism group of some geometric structure of M, more or less closely related with the Riemannian structure of M, by which M is geometrically distinguished from the other symmetric spaces. Let us observe a few examples. Let M be the sphere as a symmetric space. M has the projective [respectively, conformal] structure; it can be thought of as the set of all geodesics [respectively, the function which gives the angles between two tangent vectors at the same points] of M. This can be defined, of course, for any Riemannian manifold, but the automorphism group L, or the projective [respectively, conformal] transformation group, differs from the isometry group G for the sphere M, and by this fact, M is distinguished from all other symmetric spaces (except the real projective space which is locally isometric with M), as asserted by E. Cartan [Oeuvres completes, Partie I, Vol. II, Gauthier-Villars, Paris, 1952, p. 659]. (See [7], [8] for the proof.) M is the standard space in the projective [respectively, conformal] differential geometry. And, 'a la F. Klein, this group L on M gives rise to the (real) projective [respectively, conformal or Moebius] geometry. Next, to observe another example, we select a compact hermitian symmetric space for M. The structure is the complex structure connected with the Riemannian metric in a certain way. The automorphism group L is the holomorphic transformation group. L is a complex Lie group whose complex structure essentially determines that of M.

A Riemannian geometric invariant and its applications to a problem of Borel and Serre

Abstract: A new geometric invariant will be introduced, studied and determined on compact symmetric spaces. Introduction. We will introduce a new invariant on Riemannian manifolds, which is especially interesting on compact symmetric spaces, and we will determine the invariant for the compact symmetric spaces, thus amplifying the announcement [CN1]. A symmetric space M is defined with the point symmetry sx at every point x of M. Our new invariant, denoted by #jM, may be defined as the maximal possible cardinality #A2 of a subset A2 of M such that the point symmetry sx fixes every point of A2 for every x in A2. \"The 2-number\" #2Af is finite. #2M is clearly equal to 1 if M is not compact (but connected and simple). We thus consider compact spaces M only. When M is connected, the definition is equivalent to say that #2M is the maximal possible cardinality #A2 of a subset A2 of M such that for every pair of points, x and y, of A2 there exists a closed geodesic of M on which x and y are antipodal to each other. Thus the invariant could be defined on any connected Riemannian manifold. It is easy to see (1.4) that the geometric invariant #2M is a new obstruction to the existence of a totally geodesic embeddings /: N —> M, since the existence of / clearly implies the inequality #2./V < #2M. For example, while the complex Grassmann manifold G2(C4) of the 2-dimensional subspaces of the complex vector space C4 is obviously embedded into Gs(C6) as a totally submanifold, the space G2(C4)* which one obtains by identifying every member of G2(C4) with its orthogonal complement in C4, however, cannot be totally geodesically embedded into G3(C6)*, because #2G2(C4)* = 15 > 12 = #2G3(C6)* according to (6.4). The 2-number is not an obstruction to a topological embedding; for instance, the real projective space Gy(Rn) can be topologically embedded in a sufficiently high dimensional sphere, but the 2-number #2Gy(Rn) = n (> 2) simply prohibits a totally geodesic embedding into any sphere whose 2-number is 2 regardless of dimension. Nevertheless, the invariant, #2M, has certain bearings on the topology of M in other aspects; for instance, #2M equals xM, the Euler number of M, if Af is a semisimple hermitian symmetric space (4.3); (in particular, one thus has xM > X-B for every hermitian subspace B of a semisimple hermitian symmetric space M). And Received by the editors March 11, 1985 and, in revised form, May 8, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C21, 53C35, 22E40; Secondary 53C40.

Another Report on Harmonic Maps

Abstract: (1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Geometry of Slant Submanifolds

Abstract: The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.