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.
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
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