Example of a smooth function in a manifold whose integration vanishes [closed]

Question

Let $M$ be a complete Riemannian manifold. Now for a fixed $p\in M$, is there any non-constant smooth function $u:M\rightarrow\mathbb{R}$ such that $$\int_{B_r}udV=0\ \forall 0\leq r<\infty,$$ where $B_r$ is a ball with center at $p$ and radius $r$. I can not find such an example. I know that in $\mathbb{R}$, the above equation implies that $u$ is zero, but for general case I can not find any example. Please help!

Answer 1

On the circle $x^2+y^2=1$, the function $x$ has this property for $p=(0,1)$ or $p=(0,-1)$. On the sphere $|x|^2=1$ in $\mathbb{R}^{n+1}$, every nonzero linear function restricts to a function with this property, around any point $p$ at which it vanishes. On hyperbolic space, in the ball model, any linear function on the ball has this property about $p$ the origin. Similarly for complex or quaternionic hyperbolic space. Suppose that $M$ is a Riemannian manifold and take a point $p$ of $M$. Suppose that $r$ is the injectivity radius of $M$ at $p$. Denote the exponential map as $e \colon B_r(0) \to B_r(p)$ between the balls in $T_p M$ and in $M$. Take the Euclidean volume form $\Omega_p$ on $T_pM$ and the volume form $\Omega_M$ of $M$. Let $F>0$ be the function on $B_r(p)$ so that $e^{-1*}\Omega_p=F\Omega_M$ on $B_r M$. Let $h$ be a linear function on $T_p M$. Let $k$ be a smooth rotationally symmetric function on $T_p M$ with compact support contained in the open ball $B_r(0)$. Then the function $f$ equal to $(e^{-1*}hk)F$ in $B_r(p)$ and equal to zero outside $B_r(p)$ has zero integral on any ball around $p$ in $M$, as the integral is just the Euclidean integral of $hk$ around $0$ in $B_r(0)$.