# 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  