Semiclassical Analysis of Low Lying Eigenvalues I by Simon B.

By Simon B.

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Of course the point of the above proposition is that the equivalent properties stated there are actually true. We will now prove this fact. Firstly, in order to prove ii), it is enough to do it locally in the following sense. For every m E M there exists a neighbourhood NN such that there is a function as in ii) with V replaced by V fl Nx and U by U fl N.. , by a locally finite refinement and notes that the sum of the corresponding functions fulfils the requirement. Taking Nx to be a coordinate neighbourhood of x, we therefore reduce the problem to proving the following.

Remark. We have associated to every Lie group, a Lie algebra and to every Lie group homomorphism, a Lie algebra homomorphism of the corresponding Lie algebras. This correspondence helps one to understand an analytical object such as a Lie group, by a purely algebraic object, namely its Lie algebra. 4. Theorem of Frobenius Suppose M is a differential manifold and we are given a subbundle E of TM. We seek to find conditions under which one can assert that at every m E M, there exists a local coordinate system (U, x) such that '9 , ...

It is easy to see that the image with the induced topology is not even locally connected. Indeed this shows that the image is not a subspace at any point. If a is irrational, it goes round and round infinitely. If it is rational, it rewinds at a finite stage. We will indicate many simpler examples as well. A figure like 6 (open at the top end) can be realized as a submanifold of R2 by mapping ll8 differentiably like Clearly this figure with the topology induced from that of R2 is not a manifold at the nodal point.

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