By V. A. Tkachenko

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**Extra info for A problem in the spectral theory of an ordinary differential operator in a complex domain**

**Example text**

The converse also holds locally. That is, a submanifold can always be written locally as the graph of a smooth function. LEMMA 2 and p0 is a point on Suppose M is a smooth i-dimensional submantfold of , and a neighborhood U of the M. There is an affine linear map L: origin and a smooth function h: R1 —+ RN_I such that (i) L(p0) is the origin. (ii) L{M}flU—{(s,y)ER1 (iii) h(0) = PROOF 0 and Dh(O) 0. First, we translate coordinates so that p0 is the origin. Give RN the coordinates (x,y) with x ER1 and yE RN_I.

This can be proved by using the chain rule. , = There is a corresponding formula that relates the exterior derivative of a higher degree form in terms of its action on a wedge product of vector fields. The Analysis on Eucll4ean Space 14 formula is rather messy. Moreover, we shall only need this relationship for the exterior derivative of 1-forms. We present this case in the following lemma. LEMMA 3 Suppose is a smooth 1-form and L', L2 are smooth vector fields. , L' A L2) = L'{(4, L2)} — [L',L2]).

If V is a vector space of real dimension N, then dimR V ® C = 2N and As a complex vector space, V dimc V ® C = N. For shorthand, we write v v v v ® 1. In this way, V is naturally imbedded into V® C by identifying V with V ® 1. There is also a natural conjugation operator for V 0 C. This is defined by 39 40 Complexified Vectors and Forms As an example, let M be a smooth manifold of real dimension N. For p E M, C is called the complexified tangent space and T(M) ® C is called the complexified cotangent space.

### A problem in the spectral theory of an ordinary differential operator in a complex domain by V. A. Tkachenko

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