Centroid of manifold The centroid of a compact manifold is defined by the following equation:
$c(Y_a)$ is the centroid of the parametrized manifold $Y_a$ is the point in $\Bbb R^n$ whose $i^{th}$ coordinate is given by the equation $$c_i(Y_a)=\frac{1}{v(Y_a)}\int_{Y_a}\pi_i$$ where $\pi_i:\Bbb R^n\to\Bbb R$ is the $i^{th}$ projection function. Show that if $M$ is symmetric with respect to the subspace $x_i=0,$ then $c_i(M)=0.$
I do not know to proceed from the given even. Can someone please help me with this? I am new to this manifold stuff
Thanks in advance!
 A: If the space $V=Y_a$ is symmetric with respect to the $x^i=0$ subspace Then the integrals in question can be split into positive and negative half-manifolds.
$$ V_+ = \{ x \in V \ | \ x^i \geq 0 \} \ \ \ \ V_- = \{ x \in V \ | \ x^i \leq 0 \} $$
Then $\int_V \pi^i = \int_{V_+} \pi^i+\int_{V_-} \pi^i$ where
$$ \int_{V_+} \pi^i = \int \int \cdots \int x^i dx^1 dx^2 \cdots dx^n $$
thus as $V_+$ and $V_-$ are identical in all coordinates except $x^i$ it follows that $\int_{V_+} \pi^i = -\int_{V_-} \pi^i$. But, if $\text{Vol}(V)\neq 0$ we then obtain
$$c_i(V)=  \frac{1}{\text{Vol(V)}} \left( \int_{V_+} \pi^i+\int_{V_-} \pi^i\right) =0. $$
as the integrals over $V_+$ and $V_-$ cancel.
For example, if we have a three-fold in $\mathbb{R}^3$ symmetric with respect to $x^3=z=0$ then for each point $p = (a,b,c) \in V$ there exists another point $(a,b,-c) \in V$. Thus, $V = V_+ \cup V_-$ where $V_+ = \{ (a,b,c) \ | \ c  \geq 0 \}$ and $V_- = \{ (a,b,c) \ | \ c \leq 0 \}$. The intersection of $V_+$ and $V_-$ is a set of zero-volume so we do not double count by splitting the integral over $V$ into $V_+$ and $V_-$. Thus,
\begin{align} \notag 
c_3(V) &= \frac{1}{\text{Vol}(V)} \int_V \pi^3 \\
&=  \frac{1}{\text{Vol}(V)} \left[ \iiint_V z \, dx \, dy \, dz \right] \\
&=  \frac{1}{\text{Vol}(V)} \left[ \iiint_{V_+} z \, dx \, dy \, dz + \iiint_{V_-} z \, dx \, dy \, dz \right] \\
&= 0.
\end{align}
As the integral over $V_+$ and the integral over $V_-$ are equal in magnitude but opposite in sign by the supposed symmetry of $V$ with respect to the $z=0$ plane.
