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HARMONIC and BIHARMONIC MAPS at IASI Nowadays, the Theory of Harmonic Maps Between Riemannian Manifolds Is a Very Important Fiel

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HARMONIC and BIHARMONIC MAPS at IASI Nowadays, the Theory of Harmonic Maps Between Riemannian Manifolds Is a Very Important Fiel ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ \AL.I. CUZA" DIN IAS¸I (S.N.) MATEMATICA,˘ Tomul LVI, 2010, f.1 DOI: 10.2478/v10157-010-0006-8 HARMONIC AND BIHARMONIC MAPS AT IAS¸I BY A. BALMUS¸, D. FETCU and C. ONICIUC* Dedicated to the centennial anniversary since the foundation of the \A. Myller" Mathematical Seminar Abstract. We report on the achievements of the geometers from Ia¸siin the field of harmonic and biharmonic maps. Mathematics Subject Classification 2000: 53C07, 53C40, 53C42, 53C43, 58E20. Key words: harmonic and biharmonic maps, minimal and biharmonic submanifolds. Nowadays, the theory of harmonic maps between Riemannian manifolds is a very important field of Riemannian geometry. The harmonic maps ϕ :(M; g) ! (N; h) are critical points of the energy functional E which is defined on the infinit dimensional manifold of the smooth maps from M to N, Z 1 1 2 E : C (M; N) ! R;E(ϕ) = kdϕk vg: 2 M Here dϕ is the differential of the map ϕ and k·k denotes the Hilbert-Schmidt norm. The Euler-Lagrange equation associated to the energy E is given by the vanishing of the tension field τ(ϕ) = trace rdϕ and it is a second order elliptic equation, as one can see from its local expression @ϕβ @ϕσ −∆ϕα + N Γα gij = 0; βσ @xi @xj where Γ denotes the Christoffel symbols and ∆ is the Laplacian acting on functions. From here raises the deep link established by the harmonic maps *The third author was supported by PCE Grant PN II-2228 (502/2009), Romania 82 A. BALMUS¸, D. FETCU and C. ONICIUC 2 between the geometry of the domain and target manifolds, on one hand, and various problems from the theory of partial differential equations and calculus of variations, on the other. Although the notion of harmonic map was known before 1964, the paper \Harmonic mappings of Riemannian manifolds", by Eells and Sampson, imposed this new research field (see [25]). In this famous work the authors proved, amongst other very interesting results, that if (M; g) and (N; h) are two compact Riemannian manifolds such that (N; h) has non-positive sectional curvature, then the homotopy class of any map : M ! N has a harmonic representative ϕ :(M; g) ! (N; h), which is a minimum of the energy functional E. This existence result was proved by using analytic techniques. Since then many great mathematicians have contributed to the devel- opment of the harmonic maps theory from both the analytic and the geo- metric point of view: T. Aubin, P. Baird, F. Burstall, R. Caddeo, J. Jost, L. Lemaire, A. Lichnerowicz, Y. Ohnita, A. Sanini, R. Schoen, K. Uh- lenbeck, H. Urakawa, C.M. Wood, J.C. Wood, Y.L. Xin, S.T. Yau, etc.. Among the Romanian mathematicians we recall V. Balan, S. Dragomir, C. Gherghe, S. Ianu¸s, R. Pantilie, and at Ia¸siresults in this field were obtained by A. Balmu¸s,C.L. Bejan, D. Fetcu, C. Oniciuc, V. Oproiu and N. Papaghiuc. In the first paper on harmonic maps published by a geometer from Ia¸si, which appeared in 1988, Bejan established the first and second variation formulas for harmonic maps between semi-Riemannian manifolds (see [13]). Then followed the study of the harmonicity in the context of the tangent and cotangent bundles. Given a smooth map ϕ :(M; g) ! (N; h), the harmonicity of its differential dϕ : TM ! TN was investigated by Oproiu in [55], where TM and TN were endowed with certain semi-Riemannian metrics. Further, a different direction was followed by the same author in [56], where the harmonicity of 1-forms, thought of as maps from (M; g) to the cotangent bundle T ∗M, was studied. There was considered on T ∗M a semi-Riemannian metric G defined by G(θV ;!V ) = 0;G(θV ;XH ) = θ(X);G(XH ;Y H ) = c(X; Y ); where c is a symmetric (0; 2)-tensor field on M, θV is the vertical lift of the 1-form θ and XH is the horizontal lift of the vector field X. The horizontal distribution on T ∗M was the usual one. The author obtained 3 HARMONIC AND BIHARMONIC MAPS AT IAS¸I 83 Theorem ([56]). (i) If η is a geodesic 1-form and c is a harmonic symmetric (0; 2)-tensor field on M, then η :(M; g) ! (T ∗M; G) is a harmonic map. (ii) If c is the Ricci tensor field of (M; g), then η is harmonic if and only if η is a geodesic 1-form. (iii) If θ is a 1-form on M and c is the Lie derivative of g with respect to θ], then η is harmonic if η + θ is geodesic; −θ is harmonic. (iv) If (M; J; g) is a K¨ahlermanifold and η is a 1-form on M such that η] is a holomorphic vector field on M, then η is harmonic if and only if c is a harmonic tensor field. An extended chapter in the theory of harmonic maps is represented by the study of the harmonicity of vector fields ξ 2 C(TM) thought of as maps from the base manifold (M; g) to its tangent bundle (T M; G), where G is a (semi-)Riemannian metric. The first papers on this subject were those written by Nouhaud, Ishihara and Piu (see [46, 37, 60]), in which the complete lift of the metric g, or the Sasaki metric, were considered on TM. M.P. Piu also studied the harmonicity of unit vector fields thought of as maps in the unit tangent bundle T1M = fv 2 TM : g(v; v) = 1g equipped with the Sasaki metric. Following the same direction, one studied the harmonicity of vector fields and unit vector fields when the tangent bundle was endowed with the Cheeger-Gromoll metric ([47]), with metrics of natural lift type (see [12, 50]), and with metrics of Sasaki type or of complete lift type (see [48, 49]). In the last two cases the metrics were constructed by considering on TM a new horizontal distribution, different from the usual one, i.e. that induced by the Levi-Civita connection of g. The results above are, in general, quite rigid, since the vector fields are harmonic maps when they are parallel or Killing. For example, Theorem ([47]). Let ξ 2 C(TM) be a unit vector field on the compact manifold (M; g). Consider G to be the Cheeger-Gromoll metric on TM. Then ξ :(M; g) ! (T M; G) is a harmonic map if and only if ξ is parallel. Moreover, if ξ is a Killing unit vector field and (M; g) has constant sectional curvature, then ξ :(M; g) ! (T1M; G) is a harmonic map. Somehow contrary to these rigid results, the following was obtained. Theorem ([48]). For any vector field ξ 2 C(TM), a Riemannian metric G of Sasaki type on TM can be constructed such that ξ becomes a harmonic map. 84 A. BALMUS¸, D. FETCU and C. ONICIUC 4 The proof is based on the fact that if a vector field X : M ! TM is a Riemannian immersion, where TM is endowed with a Riemannian metric such that the canonical projection π : TM ! M is a Riemannian submersion, then X is a totally geodesic map, and thus harmonic. Then, for a fixed vector field ξ, the required metric G was obtained by considering 0 0 f δ0 on TM a new horizontal distribution H depending on ξ: H = span δxi = @ − j k j @ g −∇ i j @xi (Γiky +Bi (x)) @yj , where B = ξ and (x ; y ) denote the canonical coordinates on TM. We recall here that the usual horizontal distribution δ @ − j k @ H is generated by δxi = @xi Γiky @yj . The metric G has the expression G(XV ;Y V ) = g(X; Y );G(XH ;Y V ) = g(B(X);Y ); G(XH ;Y H ) = g(X; Y ) + g(B(X);B(Y )); or, equivalently, 0 0 0 G(XV ;Y V ) = g(X; Y );G(XH ;Y V ) = 0;G(XH ;Y H ) = g(X; Y ); 0 where XV , XH and XH denote the vertical lift, the horizontal lift with respect to H and the horizontal lift with respect to H0 of the vector field X. A particular interest was paid to harmonic maps between manifolds equipped with certain structures. A well known result of Lichnerowicz states that any holomorphic map with compact domain between two almost K¨ahlermanifolds is harmonic with the minimum energy in its homotopy class (see [40]). Further results concerning the theory of harmonic maps were obtained in the context of contact geometry (see, for example, [32, 34]). A notion which generalizes both complex and contact manifolds is that of framed '-manifolds, and the harmonic maps in this general setup were studied (see, for example, [26]). The case of complex Sasakian structures was also analyzed. In this context, three classes of harmonic maps between complex Sasakian manifolds were found. Theorem ([27]). Consider M and N to be two complex Sasakian ma- nifolds, (J; G; H; u; v; U; V; g) and (J 0;G0;H0; u0; v0;U 0;V 0; h) being the cor- responding structures. If ϕ : M ! N is a smooth map satisfying one of the following conditions: (i) dϕ = G0 dϕ G + H0 dϕ H, (ii) dϕ G = G0 dϕ, (iii) dϕ H = H0 dϕ, 5 HARMONIC AND BIHARMONIC MAPS AT IAS¸I 85 then ϕ is a harmonic map. As harmonic maps arise from a variational principle, an important prob- lem is the study of the second variation for the energy functional.
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