Some Research Themes of Aristide Sanini. 27 giugno 2008 Politecnico di Torino

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1 Some Research Themes of Aristide Sanini 27 giugno 2008 Politecnico di Torino 1

2 Research themes: 60!s: projective-differential geometry 70!s: Finsler spaces 70-80!s: geometry of foliations 80-90!s: harmonic maps, Gauss maps 90!s: submanifolds of Lie groups 2

3 Seminars/Technical Reports 3

4 Harmonic maps φ : (M, g) (N, g ) is harmonic if it is a critical point of the energy functional E(φ) := 1 2 tr gφ g dv g, M with dv g : volume element of M w.r. to the metric g. e(φ) := 1 2 tr gφ g is called energy density Intuitively (Eells-Lemaire): imagine M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ : M N prescribes how one applies the rubber onto the marble: E(φ) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when release but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not snap into a different shape. 4

5 Harmonic maps If φ t is a one-parameter variation of φ = φ 0 and v = dφ t dt t=0 φ 1 T N is the corresponding infinitesimal variation de(φ t ) dt t=0 = M (τ φ, v)dv g = τ φ, v, where τ φ := tr g Ddφ is the tension field of φ τ φ = 0: Euler-Lagrange eq. for the energy funct. E(φ) = φ is harmonic τ φ = 0 5

6 Harmonic maps and deformation of metrics 2... Applicazioni tra varietà riemanniane con energia critica rispetto a deformazioni metriche Rend. Mat. (7) 3 (1983), no. 1,

7 Harmonic maps and deformation of metrics Applicazioni tra varietà riemanniane con energia critica rispetto a deformazioni metriche, Rend. Mat. (7) 3 (1983), no. 1, The energy functional E(φ) := 1 2 tr gφ g dv g, M depends essentially from the metric = exam of the conditions for the energy to be stationary w.r. to deformation of metrics (1) arbitrary deformations: E is critical dim M = 2 and φ is weakly conformal or dim M > 2 and φ is constant (2) isovolumetric deformations: E is critical dim M = 2 and φ is weakly conformal or dim M > 2 and φ is either a homothetic immersion or constant 7

8 Harmonic maps and deformation of metrics Applicazioni tra varietà riemanniane con energia critica rispetto a deformazioni metriche Rend. Mat. (7) 3 (1983), no. 1,

9 Problemi variazionali conformi, Rend. Circ. Mat. Palermo (2) 41 (1992), no. 2 Given φ : (M, g) (N, g ) with energy density e(φ) := 1 2 tr gφ g Uhlenbeck introduced the m-energy functional E m (φ) := 1 (( ) 2 m/2 2 M m e ) ( φ ) dv g, m = dim M which agrees with energy for m = 2 and depends on the conformal structure of M only de(φ, g t ) t=0 dt = 1 2 Sm (φ), h, h = dg t dt t=0, where S m (φ) = ( 2 m e φ ) m/2 1 ( 2m e φ g φ g ) is the analog of the stress-energy tensor = 9

10 ( ) ( ) ( ) ( ) Problemi variazionali conformi, Rend. Circ. Mat. Palermo (2) 41 (1992), no. 2 (1) E m (φ) is critical w.r. to deformations of g φ is weakly conformal (2) computing the 2nd derivative, if φ is weakly conformal = φ is a local minimum of E m (φ) 10

11 with Renzo Caddeo, Metriche armoniche indotte da campi vettoriali, Rend. Sem. Fac. Sci. Univ. Cagliari 57 (1987), no. 2, T M tangent bundle of (M, g), with the Sasaki metric g s. A vector field ξ on M may be considered as a map ϕ ξ : (M, g) (T M, g s ) Study of conditions under which the induced metric ϕ ξ gs on M is harmonic with respect to g. (id : (M, g) (M, ϕ ξ gs ) harmonic) Such a situation is obtained if: (a) ξ is a conformal vector field and M is a surface, (b) ξ is Killing and M is locally flat, (c) ξ is Killing with constant length and M has constant curvature, dim M > 2. 11

12 Gauss maps and harmonic maps Harmonic maps among (unit) tangent bundles of a Riemannian manifold Harmonicity of generalized Gauss maps Rapporto interno, Politecnico Torino, 6/

13 Applicazioni armoniche tra i fibrati tangenti di varietà Riemanniane Given φ : (M, g) (N, g ), its differential Φ : T M T N endowed with the Sasaki metric φ is totally geodesic (Ddφ = 0) Φ is totally geodesic Φ is totally geodesic if φ is harmonic (τ(φ) = tr(ddφ) = 0) = (1) conditions for Φ to be harmonic (2) if Φ is harmonic and M is compact = φ totally geodesic 13

14 Applicazioni armoniche tra fibrati tangenti unitari if one modifies the metric of the unit tangent bundle by a constant [Jensen-Rigoli] Sasaki-like metric : then this condition becomes a relation among this constant, the sectional curvature of N and the dimension of M 14

15 Applicazioni armoniche tra fibrati tangenti unitari 15

16 Gauss maps and harmonic maps Harmonic maps among (unit) tangent bundles of a Riemannian manifold Harmonicity of generalized Gauss maps Rapporto interno, Politecnico Torino, 6/1988 The classical Gauss map γ maps any point x of a orientable surface immersed in R 3 the unit vector N x applied in O of R 3 and is therefore a mapping of M into the unit sphere S 2 for this reason also called spherical representation of M It allows to read several properties of the surface, in particular an extrinsic view of the gaussian curvature; a classical result is THEOREM. The Gauss map γ : M S 2 is conformal if and only if a) either M is a minimal surface b) or M is contained in a sphere. 16

17 Generalized Gauss maps of submanifolds of (M, g) m dimensional Riemannian manifold isometrically immersed in R n. The Gauss map in the Grassmannian maps any x M to the subspace of R n parallel to T x M, i.e., γ : M G m (n) with G m (n): Grassmannian of m-planes of R n (endowed with its canonical metric as symmetric space). The spherical Gauss map is the mapping ν : T 1 M S n 1 sending any unit normal vector to the point of S n obtained by its parallel transport to the origin of R n. (Chern-Lashof) 17

18 Relation with harmonic maps: (Chern) f : M 2 R n (orientable surface) is harmonic the Gauss map M G 2 (n) = Q n 2 (complex quadric in CP n 1 ) is antiolomorphic (Ruh-Vilms) γ : M G m (n) is harmonic H = 0 Generalization to arbitrary Riemannian submanifolds (Wood, Jensen-Rigoli) Gauss map in the Grassmannian : γ : M G m (T N) G m (T N): Grassmann bundle with a Sasaki-like metric spherical Gauss map: ν : T 1 M T 1 N (x, ξ) (f(x), ξ) 18

19 Q = (θ i ) 2 + λ 2 (ω a r ) 2 θ = canonical form ω = connection 1-form 19

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21 Review of results of Jensen-Rigoli and correction to the result in 21

22 ... For example, for the spherical Gauss map [Jensen-Rigoli] trace(a v A w ) = λ v, w with v, w normal vector fields and λ a function 22

23 Generalized Gauss maps for submanifolds of Submanifolds and Gauss maps, Riv. Mat. Univ. Parma (5) 3 (1994), no. 1, Recall Ruh -Vilms theorem: the Gauss map γ: (M, g) (G m (n), Γ) is harmonic H = 0. Study of the weaker property: τ γ im(γ) div S(γ) = 0 i α(e i, X) e i H = 0 in particular, if M compact orientable = H constant Study of surfaces in R n with τ γ im(γ): e.g.,m 2 N 3 (c) satisfying i α(e i, X) e i H = 0 with H 0 are ruled surfaces by geodesics intersecting orthogonally a plane curve L of constant curvature in N 3 (c). For c = 0 they are round cones 23

24 Gauss maps in the Heisenberg group Gauss map of a surface of the Heisenberg group, Boll. Un. Mat. Ital. B (7) 11 (1997), H 3 = 1 x z 0 1 y : x, y, z R Heisenberg group with the left invariant Riemannian metric ds 2 = dx 2 + dy 2 + (dz x dy) 2 nilpotent Lie group admitting large classes both of minimal and of constant mean curvature surfaces H 3 does not admit totally umbilical surfaces. The Gauss map γ : M 2 G 2 (T H 3 ) of M 2 H 3 is conformal M is minimal. 24

25 Gauss maps in the Heisenberg group Gauss map of a surface of the Heisenberg group, Boll. Un. Mat. Ital. B (7) 11 (1997), Characterization of a surface M with constant mean curvature having vertically harmonic Gauss map: in case M is minimal, it is a surface having the same analytical representation in R 3 as a plane parallel to the axis of revolution of H 3. in case M has non vanishing constant mean curvature, M is a round cylinder (in the above sense) with rulings parallel to the axis of revolution of H 3. vertically harmonic: the vertical component (w.r. to the submersion G 2 (T H 3 ) M) of the tension field vanishes 25

26 Gauss maps in the Heisenberg group with Paola Piu: One-parameter subgroups and minimal surfaces in the Heisenberg group, Note Mat. 18 (1998), no. 1, (1999)... 26

27 Gauss maps in the Heisenberg group with Paola Piu: One-parameter subgroups and minimal surfaces in the Heisenberg group, Note Mat. 18 (1998), no. 1, (1999) γ : S G 2 (T H 3 ) Gauss map of S H 3 S = exp ux exp vy, (u, v) R 2, 0 a c where X = 0 0 b and Y = α γ 0 0 β indep. vectors tangent to H 3 at the identity. are two lin. S is a minimal surface with γ vertically harmonic [X, Y ] = 0 (iff aβ αb = 0). 27

28 Gauss maps in the Heisenberg group with Paola Piu: One-parameter subgroups and minimal surfaces in the Heisenberg group, Note Mat. 18 (1998), no. 1, (1999) S is a minimal surface with γ harmonic [X, Y ] = 0 and the one-parameter subgroup σ(u) = exp ux either is a geodesic of H 3, or has torsion equal to zero (i.e., a 2 + b 2 c 2 = 0). Moreover, if σ(u) is not a geodesic, and has vanishing torsion, then the ruled surface S 1 generated by principal normal lines is flat along σ(u). 28