[Similar to this answer, except for change of notation and form of random vector studied]
Let ${ p \in [1, \infty) .}$ So ${ \lVert x \rVert _{p} := (\sum _{1} ^{n} \vert x _i \vert ^{p}) ^{\frac{1}{p}} }$ is a norm on ${ \mathbb{R} ^{n} }.$
For any ${ n \geq 1 ,}$ let ${ (B _{p} ^n) ^{+} }$ denote the positive part of ${ n }$ dimensional unit ball namely $${ (B ^n _p ) ^{+} := \lbrace x \in \mathbb{R} ^{n} : \text{each } x _i> 0, \lVert x \rVert _{p} < 1 \rbrace ,}$$ and let ${ (S ^{n} _p) ^{+} }$ denote the positive part of ${ n }$ dimensional unit sphere namely $${ (S ^{n} _p) ^{+} := \lbrace x \in \mathbb{R} ^{n+1} : \text{each } x _i > 0, \lVert x \rVert _{p} = 1 \rbrace. }$$
We have a "lifting" bijection $${ \phi : (B ^{n} _p) ^{+} \to (S ^{n} _p) ^{+} ,}$$ $${ (x _1, \ldots, x _{n}) \mapsto (x _1, \ldots, x _{n}, (1-x _{1} ^{p} - \ldots - x _{n} ^{p} ) ^{\frac{1}{p}} ) }.$$
We can study how bijection ${ \phi }$ maps small boxes. (Recall box notation as here).
For any fixed ${ x \in (B ^{n} _p ) ^{+} }$ and small ${ h },$ the difference ${ \phi(x+h) - \phi(x) }$ is approximately $${ (D\phi) _{x} (h) = \underbrace{\begin{pmatrix} 1 &0 &\cdots &0 \\ 0 &1 &\cdots &0 \\ \vdots &\vdots &\ddots &\vdots \\ 0 &0 &\cdots &1 \\ \partial _{1} \phi _{n+1} &\partial _{2} \phi _{n+1} &\ldots &\partial _{n} \phi _{n+1} \end{pmatrix}} _{(n+1) \times n \text{ matrix}} h }$$ where ${ \partial _{i} \phi _{n} }$ ${ = - (x _{i}) ^{p-1} (1- x _{1} ^{p} - \ldots - x _{n} ^{p}) ^{\frac{1}{p} -1} .}$
So any small box ${ x + \text{box}(\delta _{1} e _{1}, \ldots, \delta _{n} e _{n}) }$ in ${ (B _p ^{n}) ^{+} }$ (with ${ \delta _{i} }$s positive) gets approximately mapped to ${ \phi(x) + \text{box}((D\phi) _{x} (\delta _{1} e _{1}), \ldots, (D\phi) _{x} (\delta _{n} e _{n})). }$
Now change-of-density on mapping via bijection ${ \phi }$ can be studied. Say a random vector ${ Y = (Y _1, \ldots, Y _{n}) \in (B ^{n} _p ) ^{+} }$ has density ${ f _{Y} }.$ For any ${ y \in (B ^{n} _p) ^{+} }$ the event $${ \lbrace Y \in y + \text{box}(\delta _{1} e _{1}, \ldots , \delta _{n} e _{n}) \rbrace }$$ is approximately the event $${ \lbrace \phi(Y) \in \phi(y) + \text{box}((D\phi) _{y} (\delta _{1} e _{1}), \ldots, (D\phi) _{y} (\delta _{n} e _{n})) \rbrace }$$ giving $${ f _{Y}(y) \delta _{1} \ldots \delta _{n} \approx f _{\phi(Y)} (\phi(y)) \delta _{1} \ldots \delta _{n} \sqrt{\det (D\phi) _{y} ^{T} (D\phi) _{y} } }.$$ Hence the random vector ${ Z = \phi(Y) \in (S ^{n} _p) ^{+} }$ has density $${ f _{Z} (z) = \frac{1}{\sqrt{\det( M _{z} ^{T} M _{z}) }} f _{Y} (\phi ^{-1} (z)) \text{ for } z \in (S ^n _p) ^{+} },$$ where ${ M _{z} = (D\phi) _{ \phi ^{-1} (z)} }.$ To compute this density,
Lemma 1: Let ${ y = (y _{1} , \ldots, y _{n}) \in (B ^{n} _p ) ^{+} }.$ With ${ \phi }$ as above, the matrix of inner products ${ (D\phi) _{y} ^{T} (D\phi) _{y} }$ has determinant $${ \det (D\phi) _{y} ^{T} (D\phi) _{y} = 1 + \frac{\sum _{1} ^{n} y _i ^{2(p-1)}}{(1-y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{2(1-\frac{1}{p})}} }.$$
Pf: The matrix of inner products ${ (D\phi) _{y} ^{T} (D\phi) _{y} }$ is ${ n }$ by ${ n }.$ Its ${ (i,i) }$ entry is ${ 1 + (\partial _{i} \phi _{n+1}) ^{2} },$ and its ${ (i,j) }$ entry where ${ i \neq j }$ is ${ (\partial _{i} \phi _{n+1}) (\partial _{j} \phi _{n+1}) }.$ So we have $${ (D\phi) _{y} ^{T} (D\phi) _{y} = \text{Id} _{n} + \begin{pmatrix} \partial _{1} \phi _{n+1} \\ \vdots \\ \partial _{n} \phi _{n+1} \end{pmatrix} \begin{pmatrix} \partial _{1} \phi _{n+1} &\cdots &\partial _{n} \phi _{n+1} \end{pmatrix}}.$$ It is of the form identity plus a rank ${ 1 }$ matrix (unless all ${ y _{i} }$ are ${ 0 }$). By lemma ${ 2 }$ below, its determinant is ${ 1 }$ plus trace of this rank ${ 1 }$ matrix, i.e. \begin{align*} \det (D\phi) _{y} ^{T} (D\phi) _{y} &= 1 + \sum _{1} ^{n} (\partial _{i} \phi _{n+1}) ^{2} \\ &= 1 + \sum _{1} ^{n} (y _i) ^{2(p-1)}(1-y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{2(\frac{1}{p} - 1)} , \end{align*} as needed.
Lemma 2: Let ${ A \in \mathbb{F} ^{n \times n} }$ be a rank ${ 1 }$ matrix. Then its characteristic polynomial is given by $${ f(t) = \det(tI-A) = t ^{n-1} (t - \text{tr}(A)) }.$$ Especially, ${ \text{det}(I + A) }$ ${ = (-1) ^{n} f(-1) }$ ${ = 1 + \text{tr}(A) }.$
Pf: Since ${ A }$ has kernel of dimension ${ (n-1) }$ pick a basis ${ (e _1, \ldots, e _{n-1}) }$ of ${ \text{ker}(A) }.$ Picking a ${ v \notin \text{ker}(A) }$ gives a basis ${ (e _1, \ldots, e _{n-1}; v) }$ of ${ \mathbb{F} ^{n} }.$
Now matrix of ${ A }$ w.r.t this basis has first ${ (n-1) }$ columns zero. So characteristic polynomial of ${ A }$ is ${ f(t) = t ^{n-1} (t - \lambda ) },$ where ${ \lambda }$ is bottom right entry of above matrix. So infact ${ f(t) = t ^{n-1} (t - \text{tr}(A)) },$ as needed.
So we have:
Theorem 1 [Change of density from ${ (B _p ^n) ^{+} }$ to ${ (S _p ^{n}) ^{+} }$]
Let ${ Y \in (B ^{n} _p) ^{+} }$ be a random vector with density ${ f _Y }.$ Then the vector ${ Z = \phi(Y) \in (S ^{n} _p) ^{+} }$ has density \begin{align*} f _Z (z) &= f _{Z _1, \ldots, Z _{n+1}} (z _1, \ldots, z _{n+1}) \\ &= \frac{1}{\sqrt{\det M _{z} ^{T} M _{z} }} f _{Y} (\phi ^{-1} (z)) \\ &= \left( 1 + \frac{\sum _{1} ^{n} z _i ^{2(p-1)}}{(1- z _{1} ^{p} - \ldots - z _{n} ^{p}) ^{2(1-\frac{1}{p})}} \right) ^{- \frac{1}{2}} f _{Y _1, \ldots, Y _{n}} (z _1, \ldots, z _n). \end{align*} for ${ z = (z _1, \ldots, z _{n+1}) \in (S ^{n} _p) ^{+} }.$ Equivalently, $${ f _{Z} (z) = \left( 1 + \sum _{i=1} ^{n} \left(\frac{z _i}{z _{n+1}}\right) ^{2(p-1)} \right) ^{-\frac{1}{2}} f _{Y} (z _1, \ldots, z _n) \text{, for } z \in (S _p ^n) ^{+} . }$$
Now let ${ X _1, X _2, \ldots \geq 0 }$ be independent positive random variables with ``nice" densities ${ f _1, f _2, \ldots }$ respectively. We will soon pick appropriate ${ f _i }$s. Like here but slightly different, consider the map $${ G : (0, \infty) ^{n+1} \to (B ^n _p) ^{+} \times (0, \infty) ,}$$ $${ (x _1, \ldots, x _{n+1}) \mapsto (y _1, \ldots, y _{n+1}) = \left(\frac{x _1}{(\sum _{1} ^{n} x _{i} ^{p} + x _{n+1} ^{p}) ^{\frac{1}{p}} }, \ldots, \frac{x _n}{(\sum _{1} ^{n} x _{i} ^{p} + x _{n+1} ^{p}) ^{\frac{1}{p}} } , (\sum _{1} ^{n} x _{i} ^{p} + x _{n+1} ^{p}) ^{\frac{1}{p}} \right) }$$ which can be inverted as $${ G ^{-1} : (y _1, \ldots, y _{n+1}) \mapsto (x _1, \ldots, x _{n+1}) = (y _1 y _{n+1}, \ldots, y _{n} y _{n+1}, [(y _{n+1}) ^{p} - (y _1 y _{n+1}) ^p - \ldots - (y _n y _{n+1}) ^{p}] ^{\frac{1}{p}} ) }.$$ The last coordinate is ${ x _{n+1} = (y _{n+1}) (1- y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{\frac{1}{p}} }.$
The random vector $${ (Y _1, \ldots, Y _{n+1}) = G(X _1, \ldots, X _{n+1}) \in (B ^n _p) ^{+} \times (0, \infty) }$$ has density $${ f _{Y _1, \ldots, Y _{n+1}} (y _1, \ldots, y _{n+1}) = f _{X _1, \ldots, X _{n+1}} (G ^{-1}(y)) \vert \det D(G ^{-1}) _{y} \vert }$$ that is $${ f _{Y _1, \ldots, Y _{n+1}} (y _1, \ldots, y _{n+1}) }$$ $${ = f _{1} (y _1 y _{n+1}) \ldots f _{n} (y _{n} y _{n+1}) f _{n+1} ( (y _{n+1}) (1- y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{\frac{1}{p}}) \vert \det D(G ^{-1}) _{y} \vert .}$$
Computing ${ \det D(G ^{-1}) _{y} }$ like here we get change of density $${ f _{Y _1, \ldots, Y _{n+1}} (y _1, \ldots, y _{n+1}) }$$ $${ = f _{1} (y _1 y _{n+1}) \ldots f _{n} (y _{n} y _{n+1}) f _{n+1} ( (y _{n+1}) (1- y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{\frac{1}{p}}) (y _{n+1} ^{n}) (1- y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{\frac{1}{p} -1} }.$$
Hence the vector $${ (Y _1, \ldots, Y _n) = \left(\frac{X _1}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} }, \ldots, \frac{X _n}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} } \right) \in (B ^{n} _p) ^{+} }$$ has density \begin{align*} f _{Y _1, \ldots, Y _n} (y _1, \ldots, y _n) &= \int _{0} ^{\infty} f _{Y _1, \ldots, Y _{n+1}} (y _1, \ldots, y _{n+1}) \, dy _{n+1} \\ &=(1- y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{\frac{1}{p} -1} \int _{0} ^{\infty} f _{1} (y _1 u) \ldots f _{n} (y _{n} u ) f _{n+1} ( u (1- y _{1} ^{p} - \ldots - y _{n} ^{p}) ^{\frac{1}{p}}) u ^{n} \, du \end{align*}.
So the further transformed vector $${ Z =\phi(Y _1, \ldots, Y _n) \in (S ^{n} _p) ^{+}, }$$ that is $${ Z = \left(\frac{X _1}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} }, \ldots, \frac{X _n}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} } , \frac{X _{n+1}}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} } \right) \in (S ^n _p) ^{+} ,}$$ has density \begin{align*} f _{Z} (z ) &= f _{Z _1, \ldots, Z _{n+1}} (z _1, \ldots, z _{n+1}) \\ &= \left( 1 + \frac{\sum _{1} ^{n} z _i ^{2(p-1)}}{(1- z _{1} ^{p} - \ldots - z _{n} ^{p}) ^{2(1-\frac{1}{p})}} \right) ^{- \frac{1}{2}} f _{Y _1, \ldots, Y _{n}} (z _1, \ldots, z _n) \\ &= \left( 1 + \frac{\sum _{1} ^{n} z _i ^{2(p-1)}}{(1- z _{1} ^{p} - \ldots - z _{n} ^{p}) ^{2(1-\frac{1}{p})}} \right) ^{- \frac{1}{2}} (1-z _{1} ^{p} - \ldots - z _{n} ^{p}) ^{\frac{1}{p}-1} \\ &\, \int _{0} ^{\infty} f _{1} (z _1 u) \ldots f _{n} (z _{n} u ) f _{n+1} ( u (1- z _{1} ^{p} - \ldots - z _{n} ^{p}) ^{\frac{1}{p}}) u ^{n} \, du \\ &= \underbrace{\left(\sum _{i=1} ^{n+1} z _i ^{2(p-1)} \right) ^{-\frac{1}{2}}} _{=: \, A(z)} \underbrace{\int _{0} ^{\infty} f _{1} (z _1 u) \ldots f _{n} (z _{n} u ) f _{n+1} ( u (1- z _{1} ^{p} - \ldots - z _{n} ^{p}) ^{\frac{1}{p}}) u ^{n} \, du} _{=: \, B(z)} \end{align*} for ${ z = (z _1, \ldots, z _{n+1}) \in (S ^{n} _p) ^{+} }.$
It is useful to see in which cases above density ${ f _{Z} (z) }$ is constant over ${ z \in (S ^{n} _p) ^{+} }.$
If densities ${ f _{1} (t), \ldots, f _{n+1} (t) }$ are all proportional to ${ e ^{-{t ^p}} }$ (for ${ t \geq 0 }$), the integral term ${ B(z) }$ is constant over ${ z \in (S ^{n} _p) ^{+} }.$ If further ${ p \in \lbrace 1,2 \rbrace },$ the sum term ${ A(z) }$ is constant over ${ z \in (S ^{n} _p) ^{+} }$. This gives:
Theorem 2 [Density of normalised vector on ${ (S _p ^n) ^{+} }$ for ${ p = 1,2 }$]
Let ${ p \in \lbrace 1, 2 \rbrace }.$ If ${ X _1, \ldots, X _{n+1} }$ are independent positive random variables, with densities proportional to ${ e ^{-t ^p} }$ (for ${ t \geq 0 }$), the random vector $${ Z = \left(\frac{X _1}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} }, \ldots, \frac{X _n}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} } , \frac{X _{n+1}}{(\sum _{1} ^{n} X _{i} ^{p} + X _{n+1} ^{p}) ^{\frac{1}{p}} } \right) \in (S ^n _p) ^{+} }$$ is uniform over ${ (S _p ^{n} ) ^{+} = \lbrace x \in \mathbb{R} ^{n+1} : \text{each } x _i > 0, \lVert x \rVert _{p} = 1 \rbrace }.$
If ${ (Z _1, \ldots, Z _{n+1}) }$ is uniform over ${ (S _p ^n) ^{+}, }$ the random vector ${ ((-1) ^{S _1} Z _1, \ldots, (-1) ^{S _{n+1}} Z _{n+1}) }$ with ${ S _i }$ independent ${ \text{Unif}\lbrace 0,1 \rbrace }$ variables is uniform over the sphere ${ S _{p} ^{n} = \lbrace x \in \mathbb{R} ^{n+1} : \lVert x \rVert _{p} = 1 \rbrace }.$ In the context of above theorem $${ ((-1) ^{S _1} Z _1, \ldots, (-1) ^{S _{n+1}} Z _{n+1}) }$$ $${ = \left(\frac{(-1) ^{S _1} X _1}{(\sum _{1} ^{n} \vert (-1) ^{S _i} X _{i} \vert ^{p} + \vert (-1) ^{S _{n+1}} X _{n+1} \vert ^{p}) ^{\frac{1}{p}} }, \ldots, \frac{(-1) ^{S _n} X _n}{(\sum _{1} ^{n} \vert (-1) ^{S _i} X _{i} \vert ^{p} + \vert (-1) ^{S _{n+1}} X _{n+1} \vert ^{p}) ^{\frac{1}{p}} } , \frac{(-1) ^{S _{n+1}} X _{n+1}}{(\sum _{1} ^{n} \vert (-1) ^{S _i} X _{i} \vert ^{p} + \vert (-1) ^{S _{n+1}} X _{n+1} \vert ^{p}) ^{\frac{1}{p}} } \right). }$$ This gives:
Theorem 3 [Density of normalised vector on ${ S _p ^n }$ for ${ p = 1,2 }$]
Let ${ p \in \lbrace 1, 2 \rbrace }.$ If ${ X _1, \ldots, X _{n+1} }$ are independent random variables, with densities proportional to ${ e ^{-\vert t \vert ^p} }$ (for ${ t \in \mathbb{R} }$), the random vector $${ Z = \left(\frac{X _1}{(\sum _{1} ^{n} \vert X _{i} \vert ^{p} + \vert X _{n+1} \vert ^{p}) ^{\frac{1}{p}} }, \ldots, \frac{X _n}{(\sum _{1} ^{n} \vert X _{i} \vert ^{p} + \vert X _{n+1} \vert ^{p}) ^{\frac{1}{p}} } , \frac{X _{n+1}}{(\sum _{1} ^{n} \vert X _{i} \vert ^{p} + \vert X _{n+1} \vert ^{p}) ^{\frac{1}{p}} } \right) \in S ^n _p }$$ is uniform over ${ S _p ^{n} = \lbrace x \in \mathbb{R} ^{n+1} : \lVert x \rVert _{p} = 1 \rbrace }.$
So random variables with density proportional to ${ e ^{-t ^2}, }$ i.e. Gaussian random variables, are geometrically significant in terms of sampling uniformly from Euclidean spheres.
