Riemanian Metric and the Hodge-Star

In elementary geometry the inner product between two vectors and is given by

In the general case, the Riemanian metric is a type tensor field which at each point of the manifold satisfies

The metric tensor can be expressed as

where the components are given by

There exists an isomorphism between and . Let be a vector field ( tensor field), and let be a one-form ( tensor field). Then the isomorphism is expressed as

where with raised indices is the inverse of with lowered indices, namely

Thus the flat and sharp oprators are easily expressible in terms of the metric

From now on the convention that we will follow will be that components with raised indices will be the components of vector fields, and the components with lowered indices will be the components of one-forms. Thus will denote the components of a vector field, and will denote the components of a one-form.

Eucledian Space

The flat metric is

Given two one-forms

their inner product is given by

The hodge-star must be such that we must have the following propreties satisfied for one-forms

Clearly

satisfies these properties.

General (Non-Eucledian) Space

In this case the symmetric bilinear form defining the inner product between two vectors

The correspond inner product between two one-forms on the other hand will be given by

and the two are inverses of each other

The inner product between two one-forms is then given by

Expanding the matrix expressions

The hodge-star must be such that

where is the normalized volume form, and in 2D it is given by the following two equivalent expressions

Since we are working with one-forms the relevant inner product will be the one between one-forms, and we will use the second expression.

The wedge product in 2D is given in components as

Now we need to look for a such that , or

Clearly one way to satify that is

and noting that , then the expression for the hodge-star is given by

which for the flat metric ( and ) reverts to our expression in Eucledian space.

To check that this expression of the hodge star is indeed indempotent (up to a sign), we apply it twice

Summary

• zero-form

  

• one-form

  

  

• two-form