I am stuck at Corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.

Corollary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}^M\geq0$ and $\text{Riem}^N\leq0$. Then $\phi$ is totally geodesic.

In its proof, authors use two results, and I am confused with them:

• One result is $\int_M \triangle|d\phi|^2v_g=0$. I guess this may follow Stoke's Theorem, but why is $\triangle|d\phi|^2v_g$ an exact form?

• Another is $<d\phi\cdot\text{Ricci}^M e_s,d\phi\cdot e_s>\geq0$. We have $\text{Ricci}^M\geq0$, but we don't have $d\phi$ is a Riemannian immersion. So how do we have the result?

Any advice is helpful. Thank you.

Update $2016/2/24~10:41~\text{China}$

I have found the first result's proof. We will have $\int_M \triangle f_2~v_g=0$ by the Green's Identity $$\int_Mf_1 \triangle f_2~v_g=\int_M<\text{grad}f_1,\text{grad}f_2>v_g-\int_{\partial M}f_1Nf_2~v_g$$, which let $f_1=1$ and $\partial M$ is empty. And this can be found in Lee's Introduction to Smooth Manifold.

I am stuck at Corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.

Corollary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}^M\geq0$ and $\text{Riem}^N\leq0$. Then $\phi$ is totally geodesic.

In its proof, authors use two results, and I am confused with them:

• One result is $\int_M \triangle|d\phi|^2v_g=0$. I guess this may follow Stoke's Theorem, but why is $\triangle|d\phi|^2v_g$ an exact form?

• Another is $\langle d\phi\cdot\text{Ricci}^M e_s,d\phi\cdot e_s\rangle\geq0$. We have $\text{Ricci}^M\geq0$, but we don't have $d\phi$ is a Riemannian immersion. So how do we have the result?

Any advice is helpful. Thank you.

Update $2016/2/24~10:41~\text{China}$

I have found the first result's proof. We will have $\int_M \triangle f_2~v_g=0$ by the Green's Identity $$\int_Mf_1 \triangle f_2~v_g=\int_M\langle\text{grad}f_1,\text{grad}f_2\rangle v_g-\int_{\partial M}f_1Nf_2~v_g$$, which let $f_1=1$ and $\partial M$ is empty. And this can be found in Lee's Introduction to Smooth Manifold.

Update $2016/2/24~14:14~\text{China}$

The $\{e_s\}$ is an orthonormal basis at the point under consideration on $M$, which can be find on page $22$.

I am stuck at corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.

Collary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}^M\geq0$ and $\text{Riem}^N\leq0$. Then $\phi$ is totally geodesic.

In its proof, authors use two results, and I am confused with them:

• One result is $\int_M \triangle|d\phi|^2v_g=0$. I guess this may follow Stoke's Theorem, but why is $\triangle|d\phi|^2v_g$ an exact form?

• Another is $<d\phi\cdot\text{Ricci}^M e_s,d\phi\cdot e_s>\geq0$. We have $\text{Ricci}^M\geq0$, but we don't have $d\phi$ is a Riemannian immersion. So how do we have the result?

Any advice is helpful. Thank you.

Update $2016/2/24~10:41~\text{China}$

I have found the first result's proof. We will have $\int_M \triangle f_2~v_g=0$ by the Green's Identity $$\int_Mf_1 \triangle f_2~v_g=\int_M<\text{grad}f_1,\text{grad}f_2>v_g-\int_{\partial M}f_1Nf_2~v_g$$, which let $f_1=1$ and $\partial M$ is empty.

Green's Identity can be found in Lee's Introduction to Smooth Manifold

I am stuck at Corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.

Corollary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}^M\geq0$ and $\text{Riem}^N\leq0$. Then $\phi$ is totally geodesic.

In its proof, authors use two results, and I am confused with them:

• One result is $\int_M \triangle|d\phi|^2v_g=0$. I guess this may follow Stoke's Theorem, but why is $\triangle|d\phi|^2v_g$ an exact form?

• Another is $<d\phi\cdot\text{Ricci}^M e_s,d\phi\cdot e_s>\geq0$. We have $\text{Ricci}^M\geq0$, but we don't have $d\phi$ is a Riemannian immersion. So how do we have the result?

Any advice is helpful. Thank you.

Update $2016/2/24~10:41~\text{China}$

I have found the first result's proof. We will have $\int_M \triangle f_2~v_g=0$ by the Green's Identity $$\int_Mf_1 \triangle f_2~v_g=\int_M<\text{grad}f_1,\text{grad}f_2>v_g-\int_{\partial M}f_1Nf_2~v_g$$, which let $f_1=1$ and $\partial M$ is empty. And this can be found in Lee's Introduction to Smooth Manifold.

I am stuck at corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.

Collary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}^M\geq0$ and $\text{Riem}^N\leq0$. Then $\phi$ is totally geodesic.

In its proof, authors use two results, and I am confused with them:

• One result is $\int_M \triangle|d\phi|^2v_g=0$. I guess this may follow Stoke's Theorem, but why is $\triangle|d\phi|^2v_g$ an exact form?

• Another is $<d\phi\cdot\text{Ricci}^M e_s,d\phi\cdot e_s>\geq0$. We have $\text{Ricci}^M\geq0$, but we don't have $d\phi$ is a Riemannian immersion. So how do we have the result?

Any advice is helpful. Thank you.

I am stuck at corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.

Collary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}^M\geq0$ and $\text{Riem}^N\leq0$. Then $\phi$ is totally geodesic.

In its proof, authors use two results, and I am confused with them:

• One result is $\int_M \triangle|d\phi|^2v_g=0$. I guess this may follow Stoke's Theorem, but why is $\triangle|d\phi|^2v_g$ an exact form?

• Another is $<d\phi\cdot\text{Ricci}^M e_s,d\phi\cdot e_s>\geq0$. We have $\text{Ricci}^M\geq0$, but we don't have $d\phi$ is a Riemannian immersion. So how do we have the result?

Any advice is helpful. Thank you.

Update $2016/2/24~10:41~\text{China}$

I have found the first result's proof. We will have $\int_M \triangle f_2~v_g=0$ by the Green's Identity $$\int_Mf_1 \triangle f_2~v_g=\int_M<\text{grad}f_1,\text{grad}f_2>v_g-\int_{\partial M}f_1Nf_2~v_g$$, which let $f_1=1$ and $\partial M$ is empty.

Green's Identity can be found in Lee's Introduction to Smooth Manifold