Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|<sub>s</sub> i.e. a positive definite real inner-product on T<sub>p</sub>M...Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|<sub>s</sub> i.e. a positive definite real inner-product on T<sub>p</sub>M for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s</sub> can always be extended to a Riemannian metric on TM from a special perspective.展开更多
The main result in this paper is the following:Theorem. Assume that W is a k-connected compact PL n-manifold with boundary, BdW is,（k-Ⅰ）-connect,k≥Ⅰ. （BdW is Ⅰ-connected for k=Ⅰ）, 0≤h≤2k, 2n-h>5 and ther...The main result in this paper is the following:Theorem. Assume that W is a k-connected compact PL n-manifold with boundary, BdW is,（k-Ⅰ）-connect,k≥Ⅰ. （BdW is Ⅰ-connected for k=Ⅰ）, 0≤h≤2k, 2n-h>5 and there exists a normal block（n-h-Ⅰ）-bundle v over W. then（1） There is neat PL embedding W→D2n-hwhich normal block bundle is isomorphic to v（?） ε.（2） There is a PL embedding W→S2n-h-1which normal block bundle is isomorphic to v. Where ε denotes the trivial block l-boundle, D2n-h={x=(x1, x2,…, x2n-h∈R2n-h||xi|≤Ⅰ} and S2n-h-Ⅰ=BdD2n-h.展开更多
文摘Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|<sub>s</sub> i.e. a positive definite real inner-product on T<sub>p</sub>M for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s</sub> can always be extended to a Riemannian metric on TM from a special perspective.
文摘The main result in this paper is the following:Theorem. Assume that W is a k-connected compact PL n-manifold with boundary, BdW is,（k-Ⅰ）-connect,k≥Ⅰ. （BdW is Ⅰ-connected for k=Ⅰ）, 0≤h≤2k, 2n-h>5 and there exists a normal block（n-h-Ⅰ）-bundle v over W. then（1） There is neat PL embedding W→D2n-hwhich normal block bundle is isomorphic to v（?） ε.（2） There is a PL embedding W→S2n-h-1which normal block bundle is isomorphic to v. Where ε denotes the trivial block l-boundle, D2n-h={x=(x1, x2,…, x2n-h∈R2n-h||xi|≤Ⅰ} and S2n-h-Ⅰ=BdD2n-h.
