The Hodge dual allows the definition of an inner product for , define . . satisfies

• 
• on positive definite metrics.

Now can define , require .

Given (M,g) with ,

so . also in the euclidean case.

, , .

properties of

• 
• so is nilpotent.
• if w is co-closed
• then w is co-exact

source psfile jl@crush.caltech.edu index Hodge_decomposition_theorem complex_Hodge_dual Hodge_dual