Introduction Energy of a G-structure Harmonic almost contact structures Harmonic almost contact structures via the intrinsic torsion J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera University of La Laguna Canary Islands, Spain July 2008 J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold For an oriented Riemannian manifold M of dimension n , given G a Lie subgroup of SO ( n ), M is said to be equipped with a G -structure , if there exists a subbundle G ( M ), with structure group G , of the oriented orthonormal frame bundle SO ( M ) . J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold For an oriented Riemannian manifold M of dimension n , given G a Lie subgroup of SO ( n ), M is said to be equipped with a G -structure , if there exists a subbundle G ( M ), with structure group G , of the oriented orthonormal frame bundle SO ( M ) . G dim M name of the G -structure U ( n ) 2 n almost Hermitian SU ( n ) 2 n special almost Hermitian U ( n ) × 1 2 n + 1 almost contact metric Sp ( n ) 4 n almost hyperHermitian Sp ( n ) Sp (1) 4 n almost quaternion Hermitian 7 G 2 -structure G 2 Spin (7) 8 Spin (7)-structure J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold For a fixed G , ’which are the best G -structures on a given Riemannian manifold ( M , �· , ·� )?’ J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold For a fixed G , ’which are the best G -structures on a given Riemannian manifold ( M , �· , ·� )?’ ( M , �· , ·� M ) , ( N , �· , ·� N ) f : M → N For M compact and oriented, the energy of f is given by: � E ( f ) = 1 � f ∗ � 2 dv 2 M � f ∗ � 2 = � f ∗ e i , f ∗ e i � N J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold � E ( f ) = 1 � f ∗ � 2 dv 2 M Tension field τ ( f ) = � ∇ e i ( f ∗ e i ) − f ∗ ∇ e i e i , ∇ is the induced connection by ∇ N on f ∗ TN where � the pullback bundle f ∗ TN = { ( m , � m ∈ M and � X ) , X ∈ T f ( m ) N } f harmonic map if and only if τ ( f ) = 0 J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds , Amer. J. Math. 86 (1964), 109-160. J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold ( M , �· , ·� ) compact and oriented, G ⊆ SO ( n ), G closed and connected G ( M ) ⊆ SO ( M ) The presence of a G -structure is equivalent to the presence of a section σ : M → SO ( M ) / G � A , B � SO ( M ) / G = � π ∗ A , π ∗ B � + � φ A , φ B � . The energy of a G -structure σ � E ( σ ) = 1 � σ ∗ � 2 dv 2 M C. M. Wood, Harmonic sections of homogeneous fibre bundles , Differential Geom. Appl. 19 (2003), 193-210 J. C. Gonz´ alez-D´ avila and FMC, Harmonic G -structures , Math. Proc. Cambridge Philos. Soc. (to appear). arXiv:math.DG/0706.0116 J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold so ( n ) = g ⊕ m , � ϕ 1 , ϕ 2 � = � ϕ 1 ( u i ) , ϕ 2 ( u i ) � ( g .ϕ )( x ) = g ϕ ( g − 1 x ) , x ∈ R n so ( M ) = g σ ⊕ m σ , G -connection : � torsion : � ξ X = � ∇ , ∇ X − ∇ X ∈ so ( M ) ξ X = ( � � ξ X ) g σ + ( � ξ X ) m σ ∇ X − (˜ X = � minimal connection of σ : ∇ G ξ X ) g σ X = ( � intrinsic torsion of σ : ξ G ξ X ) m σ = ∇ G X − ∇ X ∈ m σ J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold For a G -structure σ : intrinsic torsion ξ G ∈ T ∗ M ⊗ m σ minimal connection ∇ G , ∇ G = ∇ + ξ G , S. Salamon, Riemannian Geometry and Holonomy Groups , Pitman Research Notes in Math. Series, 201 , Longman (1989). R. L. Bryant, Metrics with expceptional holonomy , Ann. of Math. 126 (1987), 525–576. R. Cleyton and A. F. Swann, Einstein metrics via intrinsic or parallel torsion , Math. Z. 247 no. 3(2004), 513–528. J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold SO ( M ) / G as a Riemannian manifold π SO ( n ) π G SO ( M ) − → M SO ( M ) − → SO ( M ) / G T SO ( M ) / G = V ⊕ H V = π G ∗ (ker π SO ( n ) ∗ ) H = π G ∗ (ker ω ) ω : T SO ( M ) → so ( n ) is the connection one-form of the Levi Civita connection ∇ J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold π SO ( M ) / G − → M π ∗ so ( M ) = SO ( M ) × G so ( n ) = g SO ( M ) ⊕ m SO ( M ) , where g SO ( M ) = SO ( M ) × G g and m SO ( M ) = SO ( M ) × G m . A fibred metric on π ∗ so ( M ) is defined by � ( pG , ϕ m ) , ( pG , ψ m ) � = � ϕ m , ψ m � J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold � � φ | V pG : V pG → m SO ( M ) pG Any vector in V pG is given by π G ∗ p ( a ∗ p ), for some a = ( a ji ) ∈ m p )) = ( pG , a ji p ( u i ) ♭ ⊗ p ( u j )) φ | V pG ( π G ∗ p ( a ∗ p is an orthonormal frame on m ∈ M , p : R n → T m M , and u i = (0 , . . . , 1 , . . . , 0) ∈ R n . Extending φ to T pG SO ( M ) / G , by saying φ | H pG = 0, one can define � A , B � SO ( M ) / G = � π ∗ A , π ∗ B � + � φ A , φ B � π : SO ( M ) / G → M is a Riemannian submersion with totally geodesic fibres [Vilms] (Besse’s book) J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a map between Riemannian manifolds Energy of a G-structure G -structures Harmonic almost contact structures SO ( M ) / G as a Riemannian manifold � A , B � SO ( M ) / G = � π ∗ A , π ∗ B � + � φ A , φ B � π : SO ( M ) / G → M is a Riemannian submersion with totally geodesic fibres � � E ( σ ) = 1 � σ ∗ � 2 dv = n 2 Vol ( M ) + 1 � φ σ ∗ � 2 dv 2 2 M M J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a G-structure σ : M → SO ( M ) / G Energy of a G-structure the first variation formula Harmonic almost contact structures Harmonic G-structures � � E ( σ ) = 1 2 Vol ( M ) + 1 � σ ∗ � 2 dv = n � φ σ ∗ � 2 dv . 2 2 M M φ σ ∗ = − ξ G Total bending � � B ( σ ) = 1 � φ σ ∗ � 2 dv = 1 � ξ G � 2 dv 2 2 M M J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
Introduction Energy of a G-structure σ : M → SO ( M ) / G Energy of a G-structure the first variation formula Harmonic almost contact structures Harmonic G-structures t → σ t ∈ Γ ∞ ( SO ( M ) / G ) variation such that σ 0 = σ variation field m → ϕ ( m ) = d dt | t =0 σ t ( m ), ϕ ∈ Γ ∞ ( σ ∗ V ). Therefore, Γ ∞ ( σ ∗ V ) ∼ = T σ Γ ∞ ( SO ( M ) / G ) σ ∗ V ∼ = σ ∗ m SO ( M ) ∼ = m σ Then T σ Γ ∞ ( SO ( M ) / G ) ∼ = Γ ∞ ( m σ ) The first variation formula If σ is a G -structure, then, for all ϕ ∈ Γ ∞ ( m σ ) ∼ = T σ Γ ∞ ( SO ( M ) / G ), we have � � � ξ G , ∇ ϕ � dv = − � d ∗ ξ G , ϕ � dv , d E σ ( ϕ ) = − M M where ξ G is the intrinsic torsion of σ . J. C. Gonz´ alez-D´ avila and F. Mart´ ın Cabrera Harmonic almost contact structures via the intrinsic torsion
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