Hodge star

$\pi _{1}(S^{1})=?\,$

This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

Let $M$ be a Riemannian manifold in $n$ dimensions with metric $g$. The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms

$*:\Omega ^{i}(T^{*}M)\rightarrow \Omega ^{n-i}(T^{*}M)$

defined as follows: Let $\phi _{1},...,\phi _{n}$ be a local orthonormal coframe (i.e., a collection of locally defined 1-forms which are orthonormal with respect to the induced metric on the cotangent space). Then we define

$*\phi _{1}\wedge \cdots \wedge \phi _{i}=\pm \phi _{i+1}\wedge \cdots \wedge \phi _{n}$

where the plus or minus is chosen so that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_1\wedge\cdots\wedge\phi_i\wedge *(\phi_1\wedge\cdots\wedge\phi_i)}

is the volume form on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . To define the Hodge star operator for general forms, we simply extend the above definition by linearity.

Example

Give Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^2} the standard metric so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx, dy} is a coframe. Then the volume form is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx\wedge dy} . Thus,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *dx = dy }

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *dy = -dx }

and in general

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle * fdx + gdy = fdy - gdx }