The Geometry of the Sasaki Metric on the Sphere Bundles of Euclidean Atiyah Vector Bundles
- PDF / 569,738 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 91 Downloads / 221 Views
The Geometry of the Sasaki Metric on the Sphere Bundles of Euclidean Atiyah Vector Bundles Mohamed Boucetta and Hasna Essoufi Abstract. Let (M, , T M ) be a Riemannian manifold. It is well known that the Sasaki metric on T M is very rigid, but it has nice properties when restricted to T (r) M = {u ∈ T M, |u| = r}. In this paper, we consider a general situation where we replace T M by a vector bundle E −→ M endowed with a Euclidean product , E and a connection ∇E which preserves , E . We define the Sasaki metric on E and we consider its restriction h to E (r) = {a ∈ E, a, aE = r2 }. We study the Riemannian geometry of (E (r) , h) generalizing many results first obtained on T (r) M and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in Boucetta and Essoufi J Geom Phys 140:161–177, 2019). Finally, we prove that any unimodular three dimensional Lie group G carries a left invariant Riemannian metric, such that (T (1) G, h) has a positive scalar curvature. Mathematics Subject Classification. 53C25, 53D17, 53C07. Keywords. Riemannian manifolds, Sasaki metric, sphere bundles.
1. Introduction Through this paper, a Euclidean vector bundle is a vector bundle πE : E −→ M endowed with , E ∈ Γ(E ∗ ⊗E ∗ ) which is bilinear symmetric and positive definite in the restriction to each fiber. Let (M, , T M ) be a Riemannian manifold of dimension n, πE : E −→ M a vector bundle of rank m endowed with a Euclidean product , E , and a linear connection ∇E which preserves , E . Denote by K : T E −→ E the connection map of ∇E locally given by: ⎞ ⎛ ⎞ ⎛ n m m n m ⎝Zl + bi ∂xi + Zj ∂μj ⎠ = bi μj Γlij ⎠ sl , K⎝ i=1
j=1
l=1
0123456789().: V,-vol
i=1 j=1
Page 2 of 30
178
M. Boucetta and H. Essoufi
MJOM
where (x1 , . . . , xn ) is a system of local coordinates, (s1 , . . . , sm ) is a basis of local sections (xi , μj ) the associated system of coordinates on E, and m of E, l ∇E Γ s . Then: l ij ∂x sj = l=1 i
T E = ker dπE ⊕ ker K. The Sasaki metric gs on E is the Riemannian metric given by: gs (A, B) = dπE (A), dπE (B)T M + K(A), K(B)E ,
A, B ∈ Ta E.
For any r > 0, the sphere bundle of radius r is the hypersurface E (r) = a ∈ E, a, aE = r2 . They are two classes of such Euclidean vector bundles naturally associated with a Riemannian manifold. We refer to the first one as the classical case. It is the case where E = T M , , E = , T M , and ∇E is the Levi–Civita connection of (M, , T M ). The second case will be called the Atiyah Euclidean vector bundle associated with a Riemannian manifold. It has been introduced by the authors in Ref. [4]. It is defined as follows.
Let (M, , T M ) be a Riemannian manifold, so(T M ) = x∈M so(Tx M ) where so(Tx M ) is the vector space of skew-symmetric endomorphisms of Tx M and k > 0. The Levi–Civita connection ∇M of (M, , T M ) defines a connection on the vector bundle so(T M ) which we will denote in the same way and it is given, for any X ∈ Γ(T M ) a
Data Loading...
