## Hodge star operator and Signature operator

Let or and be an inner product space with a fixed orthonormal basis, be the space of form

**Hodge Star Operator**

**Lemma 1** There is a unique map s.t. for any

**PROOF**. (Uniqueness) suppose we have another map , then

for every

so for every ,i.e.

(Existence) Fix an oriented orthonormal basis , for

we define

We have and

Suppose is an oriented closed Riemannian Manifold, are forms,define the inner product by

Using integration by parts and stokes theorem,we have the following equalities:

hence we yield

**Lemma 2** The formal adjoint of is ,i.e..

**Exercise** Define ,show that

**Corollary**

**Harmonic Form and Signature**

If is an and char,then

where denote the eigenspace of

If ,,then

**Theorem 1 **

**PROOF** using corollary,we could define the following bilinear form:

Let be the eigenspace of . For ,,we have:

Hence there is a decomposation

and is positive definite on and negative definite on

Using Hodge-de Rham isomorphism

the above non degenerate bilinear form is equivalent to the intersection form:

So we have

**Signature Operator**

If ,then

Let be the complex-valued forms,define

we have and

so if we write ,where denotes the -eigenspace of ,then interchanges , due to its anti-commutitivity with , i.e.:

**Definition** is called the *signature operator*

**Theorem 2 **

**PROOF ** we have following facts:

- is elliptic,hence is elliptic,so are and . and are finite
- is self-adjoint,so
- =.so consists of harmonic forms for the eigenvectors of
- for

Using these facts,we yield:

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