Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function $$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$

Definition 2: A 1-form $\omega$ side to be harmonic if it is in kernel of Laplace operator. i.e. $\Delta\omega=(d\delta+\delta d)\omega=0$.

Question: Is there relation between two above definitions? Please give an simple example.

Update: I find some theorem in this topic:

Theorem 1. If $\omega$ is harmonic and $X$ is the dual vector field, we have that $\mathrm{div}X = 0$.

Theorem 2. If $X$ is a vector field on $(M,g)$ and $\omega(v) = g(X,v)$ is the dual 1-form, then $$\mathrm{div}X = −\delta\omega.$$

Thanks.

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function $$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$

Definition 2: A 1-form $\omega$ side to be harmonic if it is in kernel of Laplace operator. i.e. $\Delta\omega=(d\delta+\delta d)\omega=0$.

Question: Is there relation between two above definitions? Please give a simple example.

Update: I find some theorem in this topic:

Theorem 1. If $\omega$ is harmonic and $X$ is the dual vector field, we have that $\mathrm{div}X = 0$.

Theorem 2. If $X$ is a vector field on $(M,g)$ and $\omega(v) = g(X,v)$ is the dual 1-form, then $$\mathrm{div}X = −\delta\omega.$$

Thanks.

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function $$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$

Definition 2: A 1-form $\omega$ side to be harmonic if it is in kernel of Laplace operator. i.e. $\Delta\omega=(d\delta+\delta d)\omega=0$.

Question: Is there relation between two above definition? Please give an simple example.

Update: I find some theorem in this topic:

Theorem 1. If $\omega$ is harmonic and $X$ is the dual vector field, we have that $\mathrm{div}X = 0$.

Theorem 2. If $X$ is a vector field on $(M,g)$ and $\omega(v) = g(X,v)$ is the dual 1-form, then $$\mathrm{div}X = −\delta\omega.$$

Thanks.

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function $$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$

Definition 2: A 1-form $\omega$ side to be harmonic if it is in kernel of Laplace operator. i.e. $\Delta\omega=(d\delta+\delta d)\omega=0$.

Question: Is there relation between two above definitions? Please give an simple example.

Update: I find some theorem in this topic:

Theorem 1. If $\omega$ is harmonic and $X$ is the dual vector field, we have that $\mathrm{div}X = 0$.

Theorem 2. If $X$ is a vector field on $(M,g)$ and $\omega(v) = g(X,v)$ is the dual 1-form, then $$\mathrm{div}X = −\delta\omega.$$

Thanks.