Trivial connected component

If there is only one connected component, Conn(y) = 1, then y is said to be connected. A node that forms its own connected component is called a trivial connected component. 

We shall consider only locally trivial bimdles hence we will just say fiber bimdle or sometimes just bundle. In some texts one allows a slightly more general type of bundles, where the fibers may be different over various path-connected components of B. We shall not make use of this generality. 

Then the boundary dM trivially fibres over F dM) with a fibre T . The image F( dM)a) of each connectedness component dM)a of the boundary dM is a connected sum of two (n — 1)-dimensional spheres and is therefore diffeomorphic to Consequently, the boundary dM ia diffeomorphic to the disconnected sum of r copies of the manifold 5" x T", where r is the number of pairs of glued tori in a given decomposition of the manifold... 

Proof of Theorem 5.3.1 Let a G Af and let V and U be such neighbourhoods of the point x that xGV cV cU, where V is the closure of V and over U (and therefore over V and P) we have a universal covering, and the cotangent bundle is trivial. Let p L Af be a standard projection, p ( ) n (L P) = Introduce the equivalence relation between the near connectedness components Qa if Qa C P and Qp C P, where P C p (U) n (L F) and is linearly connected. FVom each equivalence class choose one element and obtain (from the condition 3) of geometric simplicity) a finite set ( i,..., ). Since the manifold is complete, any element g G Xi M) can be realized by a geodesic loop with the vertex at a point x. If a loop lies in F, then its initial data can be arbitrarily little stirred with the result that the new trajectory lying sufficiently close to the... 

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