Let $L$ be a positive function of class $C^4(\mathbb R^n/\{0\})$ with homogeneous of degree one. Introducing a matrix $g$ of elements

$\begin{equation} \label{eq:1} g_{ij}=\frac{\partial^2\left(\frac{L^2}{2}\right)}{\partial x_i\partial x_j}, \end{equation}$

Deicke ^[4] showed that the matrix $g$ is positive and the following theorem, a short and elegant proof was presented in Brickell ^[1].

Theorem 1.1  Let $\det g$ be a constant on $\mathbb R^n/\{0\}.$ Then $g$ is a constant matrix on $\mathbb R^n/\{0\}$.

Theorem 1.1 is very important in affine geometry ^{[10, 11, 13]} and Finsler geometry ^[4]. There are lots of papers introducing the history and progress of these problems, for example ^[7]. A laplacian operator and Hopf maximum principle is the key point of Deicke ^[4] 's proof. However, our method depends on the concavity of the fully nonlinear operator, we give a new method to prove more generalized operator than Theorem 1.1, for considering operator $F(g),$ which including the operator of determinant.

Theorem 1.2  Let $F(g)$ be a constant on $\mathbb R^n/\{0\},$ $F(g)$ be concave with respect to matrix $g,$ and the matrix $[F^{ij}]_{1\leq i,j\leq n}=[\frac{\partial F}{\partial g_{ij}}]_{1\leq i,j\leq n}$ be positive semi-definite. Then $g$ is a constant matrix on $\mathbb R^n/\{0\}$.

In fact

(1) If $F(g)=\log \det g,$ Theorem 1.2 is just Theorem 1.1.

(2) An interesting example of Theorem 1.2 is $F(g)=(S_k(g))^{\frac 1k},$ where $S_k(g)$ is the elementary symmetric polynomial of eigenvalues of $g.$ The concavity of $F(g)$ was from Caffarelli-Nirenberg-Spruck ^[3]. A similar Liouville problem for the $S_2$ equation was obtained in ^[2].

It is easy to see that the method of Brickell ^[1] does not apply to our Theorem 1.2.

On the other hand, there are some remarkable results for homogeneous solution to partial differential equations. Han-Nadirashvili-Yuan ^[6] proved that any homogeneous order $1$ solution to nondivergence linear elliptic equations in $\mathbb R^3$ must be linear, and Nadirashvili-Yuan ^[8] proved that any homogeneous degree other than $2$ solution to fully nonlinear elliptic equations must be "harmonic". In fact, our methods can also be used to deal with the following hessian type equations

$\begin{equation} F(D^2 u)={\rm constant}. \end{equation}$

(1.2)

More recently, Nadirashvili-Vlăduţ ^[9] obtained the following theorem.

Theorem 1.3  Let $u$ be a homogeneous order 2 real analytic function in $\mathbb R^4/\{0\}$. If $u$ is a solution of the uniformly elliptic equation $F(D^2u)=0$ in $\mathbb R^4/\{0\}$, then $u$ is a quadratic polynomial.

However, our theorem say that above theorem holds provided $F$ with some concavity/convexity property. Pingali ^[12] can show for 3-dimension, there is concave operator $G$ form $F$ without some concavity/convexity property, for example

$ F(D^2u)=\det D^2u+\Delta u $

for $\lambda_1\leq \lambda_2\leq\lambda_3$ are eigenvalues of hessian matrix $D^2u.$ Then

$ G(\lambda_1,\lambda_2, \lambda_3)=\int_{36}^{\lambda_1+\lambda_2+\lambda_3+\lambda_1\lambda_2\lambda_3}\exp(-\frac{t^2}2)dt $

has a uniformly positive gradient and is concave if $\lambda_1 > 3.$ That is to say, using our methods, there is a simple proof of Theorem 1.3 if one can construct a concave operator with respect to $F$ in Theorem 1.3.

Here we firstly list the Hopf maximum principle to be used in our proof, see for example ^[5].

Lemma 2.1  Let $u$ be a $C^2$ function which satisfies the differential inequality

$\begin{equation} Lu=a^{ij}\frac{\partial^2 u}{\partial x_i\partial x_j}+b^i\frac{\partial u}{\partial x_i}\geq 0 \end{equation}$

(2.1)

in an open domain $\Omega$, where the symmetric matrix $a^{ij}$ is locally uniformly positive definite in $\Omega$ and the coefficients $a^{ij},b^i$ are locally bounded. If $u$ takes a maximum value $M$ in $\Omega$ then $u\equiv M.$

Proof of Theorem 1.2 Differentiating this equation twice with respect to $x$

$F(g)={\rm constant},$

one has

$ F^{ij}g_{ijkl}+F^{ij,pq}g_{ijk}g_{pql}=0, $

where $F^{ij}=\frac{\partial F}{\partial g_{ij}},$ $ F^{ij, pq}=\frac{\partial^2 F}{\partial g_{ij}\partial g_{pq}},$ $g_{ijk}=\frac{\partial g_{ij}}{\partial x_k},$ and $g_{ijkl}=\frac{\partial^2 g_{ij}}{\partial x_k \partial x_l}.$

The concavity of $F(g)$ with respect to $g$ says that the matrix $J_{kl}=F^{ij}g_{ijkl}$ is positive semi-definite. In particular,

$\begin{equation} \label{hjm} F^{ij}g_{ijkk}\geq 0. \end{equation}$

(2.2)

We firstly consider (2.2) as an inequality in unit sphere $S^{n-1}$,

$\begin{equation} F^{ij}g_{ijkk}\geq 0,\quad S^{n-1}, \end{equation}$

(2.3)

that is to say using Hopf maximum principle of Lemma 2.1 and taking $\Omega=S^{n-1}$, it shows that $g_{kk}$ is constant on $S^{n-1}$, and it is so on $\mathbb R^n/\{0\}$ because $g_{kk}$ is positively homogeneous of degree zero. Then, owing to the matrix $F^{ij}g_{ijkl}$ be positive semi-definite

$ F^{ij}g_{ijkl}=0. $

(1.1)

Using Hopf maximum principle again and $g_{kl}$ is positively homogeneous of degree zero, then the matrix $g$ is constant matrix. We complete the proof of Theorem 1.2.