Using the fiber bundle M over a manifold B, we define a semi-cotangent (pull-back) bundle t x B, which has a degenerate symplectic structure. We consider lifting problem of projectable geometric objects on M to the semi-cotangent bundle. Relations between lifted objects and a degenerate symplectic structure are also presented. | Turk J Math (2014) 38: 325 – 339 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Semi-cotangent bundle and problems of lifts Furkan YILDIRIM, Arif SALIMOV∗ Department of Mathematics, Faculty of Science, Atat¨ urk University, Erzurum Turkey Received: • • Accepted: Published Online: • Printed: Abstract: Using the fiber bundle M over a manifold B, we define a semi-cotangent (pull-back) bundle t ∗ B, which has a degenerate symplectic structure. We consider lifting problem of projectable geometric objects on M to the semi-cotangent bundle. Relations between lifted objects and a degenerate symplectic structure are also presented. Key words: Vector field, complete lift, basic 1-form, semi-cotangent bundle 1. Introduction Let Mn be an n-dimensional differentiable manifold of class C ∞ and π1 : Mn → Bm the differentiable bundle determined by a submersion π1 . Suppose that (x ) = (xa , xα ), a, b, . = 1, ., n − m; α, β, . = n − m + 1, ., n; i, j, . = 1, 2, ., n is a system of local coordinates adapted to the bundle π1 : Mn → Bm , i ′ ′ where xα are coordinates in Bm , and xa are fiber coordinates of the bundle π1 : Mn → Bm . If (xa , xα ) is another system of local adapted coordinates in the bundle, then we have { ′ ′ xa = xa (xb , xβ ), ′ ′ xα = xα (xβ ). () The Jacobian of () has components ( ′ (Aij ) = ′ ∂xi ∂xj ) ( = ′ Aab 0 ′ Aaβ ′ Aα β ) . Let Tx∗ (Bm )(x = π1 (e x), x e = (xa , xα ) ∈ Mn ) be the cotangent space at a point x of Bm . If pα are components of p ∈ Tx∗ (Bm ) with respect to the natural coframe {dxα }, . p = pi dxi , then by definition the set of all points (xI ) = (xa , xα , xα ) , xα = pα , α = α + m , I = 1, ., n + m is a semi-cotangent bundle t∗ (Bm ) over the manifold Mn . The semi-cotangent bundle t∗ (Bm ) has the natural bundle structure over Bm , its bundle projection π : .
