∎

11institutetext: Department of Mathematics 22institutetext: Indian Institute
of Space Science and Technology
Thiruvanathapuram, Kerala, India-695547
22email: maheshtv.16@res.iist.ac.in(Mahesh T V)
22email<EMAIL_ADDRESS>(K S Subrahamanian Moosath)

# Conformal Submersion with Horizontal Distribution

Mahesh T V and K S Subrahamanian Moosath∗ *corresponding author

(Received: date / Accepted: date)

###### Abstract

In this article, conformal submersion with horizontal distribution of
Riemannian manifolds is defined which is a generalization of the affine
submersion with horizontal distribution. Then, a necessary condition is
obtained for the existence of a conformal submersion with horizontal
distribution. For the dual connections $\nabla$ and $\overline{\nabla}$ on
manifold $\mathbf{M}$ and $\nabla^{*}$ and $\overline{\nabla}^{*}$ on manifold
$\mathbf{B}$, we show that
$\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is a conformal
submersion with horizontal distribution if and only if
$\pi:(\mathbf{M},\overline{\nabla})\longrightarrow(\mathbf{B},\overline{\nabla^{*}})$
is a conformal submersion with horizontal distribution. Also, we obtained a
necessary and sufficient condition for $\pi\circ\sigma$ to become a geodesic
of $\mathbf{B}$ if $\sigma$ is a geodesic of $\mathbf{M}$ for
$\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ a
conformal submersion with horizontal distribution.

###### Keywords:

Conformal Submersion Fundamental equation Geodesics.

###### MSC:

MSC 53A15 MSC 53C05

##  1 Introduction

Riemannian submersion is a special tool in differential geometry and it has
got application in different areas such as Kaluza-Klein theory, Yang-Mills
theory, supergravity and superstring theories, statistical machine learning
processes, medical imaging, theory of robotics and the statistical analysis on
manifolds. O’Neill o1966fundamental  defined a Riemannian submersion and
obtained the fundamental equations of Riemannian submersion for Riemannian
manifolds. In o1967submersions , O’Neill compare the geodesics of $\mathbf{M}$
and $\mathbf{B}$ for a semi-Riemannian submersion
$\pi:\mathbf{M}\longrightarrow\mathbf{B}$. Abe and Hasegawa abe2001affine
defined an affine submersion with horizontal distribution which is a dual
notion of affine immersion and obtained the fundamental equations. They
compare the geodesics of $\mathbf{M}$ and $\mathbf{B}$ for an affine
submersion with horizontal distribution
$\pi:\mathbf{M}\longrightarrow\mathbf{B}$.

Conformal submersion and the fundamental equations of conformal submersion
were also studied by many researchers, see lornia1993fe , gud1990har  for
example. Horizontally conformal submersion is a generalization of the
Riemannian submersion. Horizontally conformal submersion is a special
horizontally conformal map which got introduced independently by Fuglede
fuglede1978harmonic  and Ishihara ishihara1978mapping . Their study focuses on
the conformality relation between metrics on the Riemannian manifolds and
Levi-Civita connections. Here the study is on the conformal submersion between
Riemannian manifolds $\mathbf{M}$ and $\mathbf{B}$ and the conformality
relation between any two affine connections $\nabla$ and $\nabla^{*}$ (need
not necessarily be the Levi-Civita connection ) on $\mathbf{M}$ and
$\mathbf{B}$, respectively.

In section 2, relevant basic concepts are given. In section 3, the concept of
a conformal submersion with horizontal distribution is defined and a necessary
condition is obtained for the existence of such maps. Fundamental equations of
conformal submersion with horizontal distribution is given and show that
$\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is a conformal
submersion with horizontal distribution if and only if
$\pi:(\mathbf{M},\overline{\nabla})\longrightarrow(\mathbf{B},\overline{\nabla^{*}})$
is a conformal submersion with horizontal distribution. In section 4, we
obtained a necessary and sufficient condition for $\pi\circ\sigma$ to be a
geodesic of $\mathbf{B}$ when $\sigma$ is a geodesic of $\mathbf{M}$ for
$\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ a
conformal submersion with horizontal distribution. Throughout this paper, all
the objects are assumed to be smooth.

##  2 Affine submersion with horizontal distribution

In this section, the concepts like submersion with horizontal distribution and
affine submersion with horizontal distribution are given.

Let $\mathbf{M}$ and $\mathbf{B}$ be Riemannian manifolds with dimension $n$
and $m$ respectively with $n>m$. An onto map
$\pi:\mathbf{M}\longrightarrow\mathbf{B}$ is called a submersion if
$\pi_{*p}:T_{p}\mathbf{M}\longrightarrow T_{\pi(p)}\mathbf{B}$ is onto for all
$p\in\mathbf{M}$. For a submersion $\pi:\mathbf{M}\longrightarrow\mathbf{B}$,
$\pi^{-1}(b)$ is a submanifold of $\mathbf{M}$ of dimension $(n-m)$ for each
$b\in\mathbf{B}$. These submanifolds $\pi^{-1}(b)$ are called fibers. Set
$\mathcal{V}(\mathbf{M})_{p}$ = $Ker(\pi_{*p})$ for each $p\in\mathbf{M}$.

###### Definition 1

A submersion $\pi:\mathbf{M}\longrightarrow\mathbf{B}$ is called a submersion
with horizontal distribution if there is a smooth distribution
$p\longrightarrow\mathcal{H}(M)_{p}$ such that

$T_{p}\mathbf{M}=\mathcal{V}(\mathbf{M})_{p}\bigoplus\mathcal{H}(M)_{p}.$ (1)

We call $\mathcal{V}(\mathbf{M})_{p}$ ($\mathcal{H}(M)_{p}$) the vertical
(horizontal) subspace of $T_{p}\mathbf{M}$. $\mathcal{H}$ and $\mathcal{V}$
denote the projections of the tangent space of $\mathbf{M}$ onto the
horizontal and vertical subspaces, respectively.

###### Note 1

Let $\pi:\mathbf{M}\longrightarrow\mathbf{B}$ be a submersion with horizontal
distribution $\mathcal{H}(M)$. Then,
$\pi_{*}\mid_{\mathcal{H}(M)_{p}}:\mathcal{H}(M)_{p}\longrightarrow
T_{\pi(p)}\mathbf{B}$ is an isomorphism for each $p\in\mathbf{M}$.

###### Definition 2

A vector field $Y$ on $\mathbf{M}$ is said to be projectable if there exists a
vector field $Y_{*}$ on $\mathbf{B}$ such that $\pi_{*}(Y_{p})$ =
$Y_{*\pi(p)}$ for each $p\in\mathbf{M}$, that is $Y$ and $Y_{*}$ are $\pi$\-
related. A vector field $X$ on $\mathbf{M}$ is said to be basic if it is
projectable and horizontal. Every vector field $X$ on $\mathbf{B}$ has a
unique smooth horizontal lift, denoted by $\tilde{X}$, to $\mathbf{M}$.

###### Definition 3

Let $\nabla$ and $\nabla^{*}$ be affine connections on $\mathbf{M}$ and
$\mathbf{B}$ respectively.
$\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is said to be
an affine submersion with horizontal distribution if
$\pi:\mathbf{M}\longrightarrow\mathbf{B}$ is a submersion with horizontal
distribution and satisfies $\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})$ =
$(\nabla^{*}_{X}Y)^{\tilde{}}$, for vector fields $X,Y$ in
$\mathcal{X}(\mathbf{B})$, where $\mathcal{X}(\mathbf{B})$ denotes the set of
all vector fields on $\mathbf{B}$.

###### Note 2

Abe and Hasegawa abe2001affine  proved that the connection $\nabla$ on
$\mathbf{M}$ induces a connection $\nabla^{{}^{\prime}}$ on $\mathbf{B}$ when
$\pi:\mathbf{M}\longrightarrow\mathbf{B}$ is a submersion with horizontal
distribution and $\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})$ is projectable for
all vector fields $X$ and $Y$ on $\mathbf{B}$.

A connection $\mathcal{V}\nabla\mathcal{V}$ on the subbundle
$\mathcal{V}(\mathbf{M})$ is defined by $(\mathcal{V}\nabla\mathcal{V})_{E}V$
= $\mathcal{V}(\nabla_{E}V)$ for any vertical vector field $V$ and any vector
field $E$ on $\mathbf{M}$. For each $b\in\mathbf{B}$,
$\mathcal{V}\nabla\mathcal{V}$ induces a unique connection $\hat{\nabla}^{b}$
on the fiber $\pi^{-1}(b)$. Abe and Hasegawa abe2001affine  proved that if
$\nabla$ is torsion free, then $\hat{\nabla}^{b}$ and $\nabla^{{}^{\prime}}$
are also torsion free.

###### Definition 4

Let $\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ be an
affine submersion with horizontal distribution $\mathcal{H}(M)$. Then, the
fundamental tensors $T$ and $A$ are defined as

$\displaystyle T_{E}F$ $\displaystyle=$
$\displaystyle\mathcal{H}(\nabla_{\mathcal{V}E}\mathcal{V}F)+\mathcal{V}(\nabla_{\mathcal{V}E}\mathcal{H}F),$
(2) $\displaystyle A_{E}F$ $\displaystyle=$
$\displaystyle\mathcal{V}(\nabla_{\mathcal{H}E}\mathcal{H}F)+\mathcal{H}(\nabla_{\mathcal{H}E}\mathcal{V}F),$
(3)

for $E$ and $F$ in $\mathcal{X}(\mathbf{M})$.

###### Remark 1

For a Riemannian manifold $(\mathbf{M},g)$ with an affine connection $\nabla$
one can define the dual connection $\overline{\nabla}$ by
$Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,\overline{\nabla}_{X}Z)$, for $X,Y$ and $Z$ in
$\mathcal{X}(\mathbf{M})$. The fundamental tensors correspond to the dual
connection $\overline{\nabla}$ is denoted by $\overline{T}$ and
$\overline{A}$.

Note that these are $(1,2)$-tensors and these tensors can be defined in a
general situation, namely, it is enough that a manifold $\mathbf{M}$ has a
splitting
$T\mathbf{M}=\mathcal{V}(\mathbf{M})\bigoplus\mathcal{H}(\mathbf{M})$. Also,
note that $T_{E}$ and $A_{E}$ reverses the horizontal and vertical subspaces
and $T_{E}=T_{\mathcal{V}E}$, $A_{E}=A_{\mathcal{H}E}$.

##  3 Conformal submersion with horizontal distribution

In this section, we generalize the concept of affine submersion with
horizontal distribution to conformal submersion with horizontal distribution
and prove a necessary condition for the existence of such maps. Then, we show
that $\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is a
conformal submersion with horizontal distribution if and only if
$\pi:(\mathbf{M},\overline{\nabla})\longrightarrow(\mathbf{B},\overline{\nabla^{*}})$
is a conformal submersion with horizontal distribution.

###### Definition 5

Let $(\mathbf{M},g_{m})$ and $(\mathbf{B},g_{b})$ be Riemannian manifolds. A
submersion $\pi:(\mathbf{M},g_{m})\longrightarrow(\mathbf{B},g_{b})$ is called
a conformal submersion if there exists a $\phi\in C^{\infty}(\mathbf{M})$ such
that

$g_{m}(X,Y)=e^{2\phi}g_{b}(\pi_{*}X,\pi_{*}Y),$ (4)

for all horizontal vector fields $X,Y\in\mathcal{X}(\mathbf{M})$.

For $\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ an affine
submersion with horizontal distribution,
$\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})=(\nabla^{*}_{X}Y)^{\tilde{}}$, for
$X,Y\in\mathcal{X}(\mathbf{B})$. In the case of conformal submersion we prove
the following theorem, which is the motivation for us to generalize the
concept of affine submersion with horizontal distribution.

###### Theorem 3.1

Let $\pi:(\mathbf{M},g_{m})\longrightarrow(\mathbf{B},g_{b})$ be a conformal
submersion. If $\nabla$ on $\mathbf{M}$ and $\nabla^{*}$ on $B$ are the Levi-
Civita connections, then

$\displaystyle\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})$ $\displaystyle=$
$\displaystyle\tilde{(\nabla^{*}_{X}Y)}+\tilde{X}(\phi)\tilde{Y}+\tilde{Y}(\phi)\tilde{X}-\mathcal{H}(grad_{\pi}\phi)g_{m}(\tilde{X},\tilde{Y}),$

where $X,Y\in\mathcal{X}(\mathbf{B})$ and $\tilde{X},\tilde{Y}$ denote their
unique horizontal lifts on $\mathbf{M}$ and $grad_{\pi}\phi$ denote the
gradient of $\phi$ with respect to $g_{m}$

###### Proof

We have the Koszul formula for the Levi-Civita connection,

$\displaystyle 2g_{m}(\nabla_{\tilde{X}}\tilde{Y},\tilde{Z})$ $\displaystyle=$
$\displaystyle\tilde{X}g_{m}(\tilde{Y},\tilde{Z})+\tilde{Y}g_{m}(\tilde{Z},\tilde{X})-\tilde{Z}g_{m}(\tilde{X},\tilde{Y})-g_{m}(\tilde{X},[\tilde{Y},\tilde{Z}])$
(5)
$\displaystyle+g_{m}(\tilde{Y},[\tilde{Z},\tilde{X}])+g_{m}(\tilde{Z},[\tilde{X},\tilde{Y}]).$

Now, consider

$\displaystyle\tilde{X}g_{m}(\tilde{Y},\tilde{Z})$ $\displaystyle=$
$\displaystyle\tilde{X}(e^{2\phi}g_{b}(Y,Z))$ $\displaystyle=$
$\displaystyle\tilde{X}(e^{2\phi})g_{b}(Y,Z)+e^{2\phi}\tilde{X}(g_{b}(Y,Z))$
$\displaystyle=$ $\displaystyle
2e^{2\phi}\tilde{X}(\phi)g_{b}(Y,Z)+e^{2\phi}Xg_{b}(Y,Z).$

Similarly,

$\displaystyle\tilde{Y}g_{m}(\tilde{X},\tilde{Z})$ $\displaystyle=$
$\displaystyle 2e^{2\phi}\tilde{Y}(\phi)g_{b}(X,Z)+e^{2\phi}Yg_{b}(X,Z).$
$\displaystyle\tilde{Z}g_{m}(\tilde{X},\tilde{Y})$ $\displaystyle=$
$\displaystyle 2e^{2\phi}\tilde{Z}(\phi)g_{b}(X,Y)+e^{2\phi}Zg_{b}(X,Y).$

Also,

$\displaystyle g_{m}(\tilde{X},[\tilde{Y},\tilde{Z}])$ $\displaystyle=$
$\displaystyle e^{2\phi}g_{b}(X,[Y,Z]).$ $\displaystyle
g_{m}(\tilde{Y},[\tilde{Z},\tilde{X}])$ $\displaystyle=$ $\displaystyle
e^{2\phi}g_{b}(Y,[Z,X]).$ $\displaystyle
g_{m}(\tilde{Z},[\tilde{X},\tilde{Y}])$ $\displaystyle=$ $\displaystyle
e^{2\phi}g_{b}(Z,[X,Y]).$

Then, from the equation (5) and the above equations

$\displaystyle 2g_{m}(\nabla_{\tilde{X}}\tilde{Y},\tilde{Z})$ $\displaystyle=$
$\displaystyle
2\tilde{X}(\phi)e^{2\phi}g_{b}(Y,Z)+2\tilde{Y}(\phi)e^{2\phi}g_{b}(X,Z)-2\tilde{Z}(\phi)e^{2\phi}g_{b}(X,Y)$
$\displaystyle+2e^{2\phi}g_{b}(\nabla^{*}_{X}Y,Z).$

Since, $\tilde{Z}(\phi)$ = $e^{2\phi}g_{b}(\pi_{*}(grad_{\pi}\phi),Z)$ we get

$\displaystyle\pi_{*}(\nabla_{\tilde{X}}\tilde{Y})$ $\displaystyle=$
$\displaystyle\nabla^{*}_{X}Y+\tilde{X}(\phi)Y+\tilde{Y}(\phi)X-e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(X,Y).$

Hence,

$\displaystyle\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})$ $\displaystyle=$
$\displaystyle\tilde{(\nabla^{*}_{X}Y)}+\tilde{X}(\phi)\tilde{Y}+\tilde{Y}(\phi)\tilde{X}-\mathcal{H}(grad_{\pi}\phi)g_{m}(\tilde{X},\tilde{Y}).$

Now, we generalize the concept of affine submersion with horizontal
distribution.

###### Definition 6

Let $\pi:(\mathbf{M},g_{m})\longrightarrow(\mathbf{B},g_{b})$ be a conformal
submersion and let $\nabla$ and $\nabla^{*}$ be affine connections on
$\mathbf{M}$ and $\mathbf{B}$, respectively. Then,
$\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is said to be
a conformal submersion with horizontal distribution
$\mathcal{H}(\mathbf{M})=\mathcal{V}(\mathbf{M})^{\perp}$ if

$\displaystyle\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})$ $\displaystyle=$
$\displaystyle\tilde{(\nabla^{*}_{X}Y)}+\tilde{X}(\phi)\tilde{Y}+\tilde{Y}(\phi)\tilde{X}-\mathcal{H}(grad_{\pi}\phi)g_{m}(\tilde{X},\tilde{Y}),$

for some $\phi\in C^{\infty}(\mathbf{M})$ and for all
$X,Y\in\mathcal{X}(\mathbf{B})$.

###### Note 3

If $\phi$ is constant, it turns out to be an affine submersion with horizontal
distribution.

###### Example 1

Let $H^{n}=\\{(x_{1},...,x_{n})\in\mathbf{R}^{n}:x_{n}>0\\}$ and
$\tilde{g}=\frac{1}{x_{n}^{2}}g$ be a Riemannian metric on $H^{n}$, where $g$
is the Euclidean metric on $\mathbf{R}^{n}$. Let
$\pi:H^{n}\longrightarrow\mathbf{R}^{n-1}$ be defined by
$\pi(x_{1},...,x_{n})=(x_{1},...,x_{n-1}).$ Let
$\phi:H^{n}\longrightarrow\mathbf{R}$ be defined by
$\phi(x_{1},...,x_{n})=\log(\frac{1}{x_{n}^{2}})$. Then, we have

$\displaystyle\tilde{g}\left(\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial
x_{j}}\right)=e^{\phi}g\left(\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial x_{j}}\right).$

Hence, $\pi:(H^{n},\tilde{g})\longrightarrow(\mathbf{R}^{n-1},g)$ is a
conformal submersion. Then, by the theorem (3.1),
$\pi:(H^{n},\nabla)\longrightarrow(\mathbf{R}^{n-1},\nabla^{*})$ is a
conformal submersion with horizontal distribution, where $\nabla$ and
$\nabla^{*}$ are Levi-Civita connections on $H^{n}$ and $\mathbf{R}^{n-1}$,
respectively.

Now, we prove a necessary condition for the existence of a conformal
submersion with horizontal distribution.

###### Theorem 3.2

Let $\pi:(\mathbf{M},g_{m})\longrightarrow(\mathbf{B},g_{b})$ be a conformal
submersion and $\nabla$ be an affine connection on $\mathbf{M}$. Assume that
$\pi:\mathbf{M}\longrightarrow\mathbf{B}$ is a submersion with horizontal
distribution. If $\mathcal{H}(\nabla_{\tilde{X}}\tilde{Y})$ is projectable for
all vector fields $X$ and $Y$ on $\mathbf{B}$. Then, there exists a unique
connection $\nabla^{*}$ on $\mathbf{B}$ such that
$\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is a conformal
submersion with horizontal distribution.

###### Proof

Setting $\nabla^{*}_{X}Y$ =
$\pi_{*}(\nabla_{\tilde{X}}\tilde{Y})-\tilde{X}(\phi)Y-\tilde{Y}(\phi)X+e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(X,Y)$,
we show that $\nabla^{*}$ is an affine connection on $\mathbf{B}$. For
example, since $\tilde{(fX)}=(f\circ\pi)\tilde{X}$, we have

$\displaystyle\nabla^{*}_{X}(fY)$ $\displaystyle=$
$\displaystyle\pi_{*}(\nabla_{\tilde{X}}(f\circ\pi)\tilde{Y})-\tilde{X}(\phi)fY-\tilde{(fY)}(\phi)X$
(6) $\displaystyle+e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(X,fY).$

Now, consider

$\displaystyle\pi_{*}(\nabla_{\tilde{X}}(f\circ\pi)\tilde{Y})$
$\displaystyle=$
$\displaystyle\pi_{*}((\tilde{X}(f\circ\pi))\tilde{Y}+(f\circ\pi)\nabla_{\tilde{X}}\tilde{Y})$
(7) $\displaystyle=$ $\displaystyle
X(f)Y+f\nabla^{*}_{X}Y+\tilde{X}(\phi)fY+f\tilde{Y}(\phi)X$
$\displaystyle-e^{2\phi}\pi_{*}(grad_{\pi}\phi)fg_{b}(X,Y).$

Then, from the equations (6) and (7)

$\displaystyle\nabla^{*}_{X}(fY)$ $\displaystyle=$ $\displaystyle
X(f)Y+f\nabla^{*}_{X}Y.$

The other condition for affine connection can be proved similarly. The
uniqueness is clear from the definition.

As in the case of affine submersion with horizontal distribution we have,

###### Lemma 1

Let $\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ be
conformal submersion with horizontal distribution, then

$\displaystyle\mathcal{H}(Tor(\nabla)(\tilde{X},\tilde{Y}))$ $\displaystyle=$
$\displaystyle\widetilde{(Tor(\nabla^{*})(X,Y))}.$ (8)
$\displaystyle\mathcal{V}(Tor(\nabla)(V,W))$ $\displaystyle=$
$\displaystyle(Tor(\hat{\nabla})(V,W)).$ (9)

###### Proof

Proof follows immediately form the definition of the conformal submersion with
horizontal distribution.

###### Corollary 1

If $\nabla$ is torsion-free, then $\nabla^{*}$ and $\hat{\nabla}$ are also
torsion-free.

Fundamental tensors $T$ and $A$ for a conformal submersion with horizontal
distribution $\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$
are defined for $E$ and $F$ in $\mathcal{X}(\mathbf{M})$ by

$T_{E}F=\mathcal{H}\nabla_{\mathcal{V}E}(\mathcal{V}F)+\mathcal{V}\nabla_{\mathcal{V}E}(\mathcal{H}F).$

and

$A_{E}F=\mathcal{V}\nabla_{\mathcal{H}E}(\mathcal{H}F)+\mathcal{H}\nabla_{\mathcal{H}E}(\mathcal{V}F).$

Note that these are $(1,2)$-tensors.

We have the fundamental equations correspond to the conformal submersion with
horizontal distribution. Let $R$ be the curvature tensor of
$(\mathbf{M},\nabla)$ defined by

$\displaystyle
R(E,F)G=\nabla_{[X,Y]}G-\nabla_{E}\nabla_{F}G+\nabla_{F}\nabla_{E}G,$

for $E,F$ and $G$ in $\mathcal{X}(\mathbf{M})$. Similarly, we denote the
curvature tensor of $\nabla^{*}$ (respectively $\hat{\nabla}$) by $R^{*}$
(respectively $\hat{R}$). Define $(1,3)$-tensors $R^{P_{1},P_{2},P_{3}}$ for
conformal submersion with horizontal distribution by

$\displaystyle R^{P_{1},P_{2},P_{3}}(E,F)G$ $\displaystyle=$ $\displaystyle
P_{3}\nabla_{[P_{1}E,P_{2}F]}P_{3}G-P_{3}\nabla_{P_{1}E}(P_{3}\nabla_{P_{2}F}P_{3}G)$
$\displaystyle+P_{3}\nabla_{P_{2}F}(P_{3}\nabla_{{P_{1}E}}P_{3}G),$

where $P_{i}=\mathcal{H}$ or $\mathcal{V}$($i=1,2,3$)and $E,F,G$ are in
$\mathcal{X}(\mathbf{M})$. Then, the following fundamental equations for
conformal submersion with horizontal distribution can be obtained.

###### Theorem 3.3

Let $X,Y,Z$ be horizontal and $U,V,W$ vertical vector fields in $\mathbf{M}$.
Then,

$\displaystyle\mathcal{V}R(U,V)W$ $\displaystyle=$ $\displaystyle
R^{\mathcal{V}\mathcal{V}\mathcal{V}}(U,V)W+T_{V}T_{U}W-T_{U}T_{V}W.$
$\displaystyle\mathcal{H}R(U,V)W$ $\displaystyle=$
$\displaystyle\mathcal{H}(\nabla_{V}T)_{U}W-\mathcal{H}(\nabla_{U}T)_{V}W-T_{Tor(\nabla)(U,V)}W.$
$\displaystyle\mathcal{V}R(U,V)X$ $\displaystyle=$
$\displaystyle\mathcal{H}(\nabla_{V}T)_{U}X-\mathcal{V}(\nabla_{U}T)_{V}X-T_{Tor(\nabla)(U,V)}X.$
$\displaystyle\mathcal{H}R(U,V)X$ $\displaystyle=$ $\displaystyle
R^{\mathcal{V}\mathcal{V}\mathcal{H}}(U,V)X+T_{V}T_{U}X-T_{U}T_{V}X.$
$\displaystyle\mathcal{V}R(U,X)V$ $\displaystyle=$ $\displaystyle
R^{\mathcal{V}\mathcal{H}\mathcal{V}}(U,X)V-T_{U}A_{X}V-A_{X}T_{U}V.$
$\displaystyle\mathcal{H}R(U,X)V$ $\displaystyle=$
$\displaystyle\mathcal{H}(\nabla_{X}T)_{U}V-\mathcal{H}(\nabla_{U}A)_{X}V-A_{A_{X}U}V+T_{T_{U}X}V$
$\displaystyle-T_{Tor(\nabla)(U,X)}V-A_{Tor(\nabla)(U,X)}V.$
$\displaystyle\mathcal{V}R(U,X)Y$ $\displaystyle=$
$\displaystyle\mathcal{V}(\nabla_{X}T)_{U}Y-\mathcal{V}(\nabla_{U}A)_{X}Y-A_{A_{X}U}Y+T_{T_{U}X}Y$
$\displaystyle-T_{Tor(\nabla)(U,X)}Y-A_{Tor(\nabla)(U,X)}Y.$
$\displaystyle\mathcal{H}R(U,X)Y$ $\displaystyle=$ $\displaystyle
R^{\mathcal{V}\mathcal{H}\mathcal{H}}(U,X)Y-T_{V}A_{X}Y+A_{X}T_{U}Y.$
$\displaystyle\mathcal{V}R(X,Y)U$ $\displaystyle=$ $\displaystyle
R^{\mathcal{H}\mathcal{H}\mathcal{V}}(X,Y)U+A_{Y}A_{X}U-A_{X}A_{Y}U.$
$\displaystyle\mathcal{H}R(X,Y)U$ $\displaystyle=$
$\displaystyle\mathcal{H}(\nabla_{Y}A)_{X}U-\mathcal{H}(\nabla_{X}A)_{Y}U+T_{A_{X}Y}U-T_{A_{Y}X}U$
$\displaystyle-T_{Tor(\nabla)(X,Y)}U-A_{Tor(\nabla)(X,Y)}U.$
$\displaystyle\mathcal{V}R(X,Y)Z$ $\displaystyle=$
$\displaystyle\mathcal{V}(\nabla_{Y}A)_{X}Z-\mathcal{V}(\nabla_{X}A)_{Y}Z+T_{A_{X}Y}Z-T_{A_{Y}X}Z$
$\displaystyle-T_{Tor(\nabla)(X,Y)}Z-A_{Tor(\nabla)(X,Y)}Z.$
$\displaystyle\mathcal{H}R(X,Y)Z$ $\displaystyle=$ $\displaystyle
R^{\mathcal{H}\mathcal{H}\mathcal{H}}(X,Y)Z+A_{Y}A_{X}Z-A_{X}A_{Y}Z.$

Now, for semi-Riemannian manifolds $(\mathbf{M},g_{m})$, $(\mathbf{B},g_{b})$
with affine connections $\nabla$ and $\nabla^{*}$ and the dual connections
$\overline{\nabla}$ and $\overline{\nabla^{*}}$ respectively we prove

###### Proposition 1

Let $\pi:(\mathbf{M},g_{m})\longrightarrow(\mathbf{B},g_{b})$ be a conformal
submersion. Then,
$\pi:(\mathbf{M},\nabla)\longrightarrow(\mathbf{B},\nabla^{*})$ is a conformal
submersion with horizontal distribution if and only if
$\pi:(\mathbf{M},\overline{\nabla})\longrightarrow(\mathbf{B},\overline{\nabla^{*}})$
is a conformal submersion with horizontal distribution.

###### Proof

Consider,

$\displaystyle\tilde{X}g_{m}(\tilde{Y},\tilde{Z})$ $\displaystyle=$
$\displaystyle 2e^{2\phi}\tilde{X}(\phi)g_{b}(Y,Z)+e^{2\phi}Xg_{b}(Y,Z)$ (10)
$\displaystyle=$ $\displaystyle
2e^{2\phi}\tilde{X}(\phi)g_{b}(Y,Z)+e^{2\phi}\\{g_{b}(\nabla^{*}_{X}Y,Z)+g_{b}(Y,\overline{\nabla^{*}}_{X}Z)\\}.$

Now consider

$\displaystyle\tilde{X}g_{m}(\tilde{Y},\tilde{Z})$ $\displaystyle=$
$\displaystyle
g_{m}(\nabla_{\tilde{X}}\tilde{Y},\tilde{Z})+g_{m}(\tilde{Y},\overline{\nabla}_{\tilde{X}}\tilde{Z})$
(11) $\displaystyle=$ $\displaystyle
e^{2\phi}g_{b}(\pi_{*}(\nabla_{\tilde{X}}\tilde{Y}),Z)+e^{2\phi}g_{b}(Y,\pi_{*}(\overline{\nabla}_{\tilde{X}}\tilde{Z})).$

Since,

$\displaystyle\pi_{*}(\nabla_{\tilde{X}}\tilde{Y})$ $\displaystyle=$
$\displaystyle\nabla^{*}_{X}Y+\tilde{X}(\phi)Y+\tilde{Y}(\phi)X-e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(X,Y).$
(12)

from the equations (10),(11) and (12) we get

$\displaystyle\pi_{*}(\overline{\nabla}_{\tilde{X}}\tilde{Z})$
$\displaystyle=$
$\displaystyle\overline{\nabla^{*}}_{X}Z+\tilde{X}(\phi)Z+\tilde{Z}(\phi)X-\pi_{*}(grad_{\pi}\phi)e^{2\phi}g_{b}(X,Z).$

Hence,
$\pi:(\mathbf{M},\overline{\nabla})\longrightarrow(\mathbf{B},\overline{\nabla^{*}})$
is a conformal submersion with horizontal distribution.
Converse is obtained by interchanging $\nabla$, $\nabla^{*}$ with
$\overline{\nabla}$, $\overline{\nabla^{*}}$ in the above proof.

##  4 Geodesics

In this section, for a conformal submersion with horizontal distribution we
prove a necessary and sufficient condition for $\pi\circ\sigma$ to be a
geodesic of $\mathbf{B}$ when $\sigma$ is a geodesic of $\mathbf{M}$. Let $M$,
$B$ be Riemannian manifolds and $\pi:M\rightarrow B$ be a submersion. Let $E$
be a vector field on a curve $\sigma$ in $\mathbf{M}$ and the horizontal part
$\mathcal{H}(E)$ and the vertical part $\mathcal{V}(E)$ of $E$ be denoted by
$H$ and $V$, respectively. $\pi\circ\sigma$ is a curve in $B$ and $E_{*}$
denote the vector field $\pi_{*}(E)=\pi_{*}(H)$ on the curve $\pi\circ\sigma$
in $\mathbf{B}$. $E_{*}^{{}^{\prime}}$ denote the covariant derivative of
$E_{*}$ and is a vector field on $\pi\circ\sigma$. The horizontal lift to
$\sigma$ of $E_{*}^{{}^{\prime}}$ is denoted by $\tilde{E_{*}^{{}^{\prime}}}$.
In o1967submersions , O’Neil compared the geodesics for semi-Riemannian
submersion and Abe and Hasegawa abe2001affine  have done it for affine
submersion with horizontal distribution.

Let $\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ be
a conformal submersion with horizontal distribution $\mathcal{H}(\mathbf{M})$.
Throughout this section we assume $\nabla$ is torsion free. A curve $\sigma$
is a geodesic if and only if $\mathcal{H}(\sigma^{{}^{\prime\prime}})=0$ and
$\mathcal{V}(\sigma^{{}^{\prime\prime}})=0$, where
$\sigma^{{}^{\prime\prime}}$ is the covariant derivative of
$\sigma^{{}^{\prime}}$. So, first we obtain the equations for
$\mathcal{H}(E^{{}^{\prime}})$ and $\mathcal{V}(E^{{}^{\prime}})$ for a vector
field $E$ on the curve $\sigma$ in $\mathbf{M}$ for a conformal submersion
with horizontal distribution.

###### Theorem 4.1

Let $\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ be
a conformal submersion with horizontal distribution, and let $E=H+V$ be a
vector field on curve $\sigma$ in $\mathbf{M}$. Then, we have

$\displaystyle\pi_{*}(\mathcal{H}(E^{{}^{\prime}}))$ $\displaystyle=$
$\displaystyle
E_{*}^{{}^{\prime}}+\pi_{*}(A_{X}U+A_{X}V+T_{U}V)-e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(\pi_{*}X,\pi_{*}H)$
$\displaystyle+X(\phi)\pi_{*}H+H(\phi)\pi_{*}X,$
$\displaystyle\mathcal{V}(E^{{}^{\prime}})$ $\displaystyle=$ $\displaystyle
A_{X}H+T_{U}H+\mathcal{V}(V^{{}^{\prime}}),$

where $X=\mathcal{H}(\sigma^{{}^{\prime}})$ and
$U=\mathcal{V}(\sigma^{{}^{\prime}})$.

###### Proof

Consider a neighborhood of an arbitrary point $\sigma(t)$ of the curve
$\sigma$ in $\mathbf{M}$. By choosing the base fields $W_{1},....,W_{n}$,
where $n=dim\mathbf{B}$, near $\pi(\sigma(t))$ on $\mathbf{B}$ and an
appropriate vertical base field near $\sigma(t)$, we can derive

$\displaystyle(E_{*}^{{}^{\prime}})_{t}$ $\displaystyle=$
$\displaystyle\sum_{i}r^{i^{\prime}}(t)(W_{i})_{\pi(\sigma(t))}+\sum_{i,k}r^{i}(t)s^{k}(t)(\nabla^{*}_{W_{k}}W_{i})_{\pi(\sigma(t))},$
(13) $\displaystyle\pi_{*}(\mathcal{H}(E^{{}^{\prime}})_{t})$ $\displaystyle=$
$\displaystyle\sum_{i}r^{i^{\prime}}(t)(W_{i})_{\pi(\sigma(t))}+\sum_{i,k}r^{i}(t)s^{k}(t)\pi_{*}(\mathcal{H}(\nabla_{\tilde{W_{k}}}\tilde{W_{i}}))_{\pi(\sigma(t))}$
(14) $\displaystyle+\pi_{*}((A_{H}U)+(A_{X}V)+(T_{U}V))_{\pi(\sigma(t))},$

where $\tilde{W}_{i}$ be the horizontal lift of $W_{i}$, for $i=1,2,...n$ and
$r^{i}(t)$ ( respectively $s^{k}(t)$) be the coefficients of $H$ (respectively
of $X$ ) in the representation using the base fields $\tilde{W}_{i}$
restricted to $\sigma$.

Since $\pi$ is a conformal submersion with horizontal distribution

$\displaystyle\pi_{*}(\mathcal{H}(\nabla_{H}X))=\nabla^{*}_{\pi_{*}(H)}\pi_{*}X+X(\phi)\pi_{*}H+H(\phi)\pi_{*}X-e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(\pi_{*}X,\pi_{*}H).$

Hence

$\displaystyle\pi_{*}(\mathcal{H}(E^{{}^{\prime}}))$ $\displaystyle=$
$\displaystyle
E_{*}^{{}^{\prime}}+\pi_{*}(A_{X}U+A_{X}V+T_{U}V)-e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(\pi_{*}X,\pi_{*}H)$
$\displaystyle+X(\phi)\pi_{*}H+H(\phi)\pi_{*}X.$

Similarly we can prove
$\mathcal{V}(E^{{}^{\prime}})=A_{X}H+T_{U}H+\mathcal{V}(V^{{}^{\prime}}).$

For $\sigma^{{}^{\prime\prime}}$ we have

###### Corollary 2

Let $\sigma$ be a curve in $\mathbf{M}$ with
$X=\mathcal{H}(\sigma^{{}^{\prime}})$ and
$U=\mathcal{V}(\sigma^{{}^{\prime}})$. Then,

$\displaystyle\pi_{*}(\mathcal{H}(\sigma^{{}^{\prime\prime}}))$
$\displaystyle=$
$\displaystyle\sigma_{*}^{{}^{\prime\prime}}+\pi_{*}(2A_{X}U+T_{U}U)-e^{2\phi}\pi_{*}(grad_{\pi}\phi)g_{b}(\pi_{*}X,\pi_{*}X)$
(15) $\displaystyle+2X(\phi)\pi_{*}X,$
$\displaystyle\mathcal{V}(\sigma^{{}^{\prime\prime}})$ $\displaystyle=$
$\displaystyle A_{X}X+T_{U}X+\mathcal{V}(U^{{}^{\prime}}),$ (16)

where $\sigma_{*}^{{}^{\prime\prime}}$ denotes the covariant derivative of
$(\pi\circ\sigma)^{{}^{\prime}}$.

Now for a conformal submersion with horizontal distribution we prove a
necessary and sufficient condition for $\pi\circ\sigma$ to become a geodesic
of $\mathbf{B}$ when $\sigma$ is a geodesic of $\mathbf{M}$.

###### Theorem 4.2

Let $\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ be
a conformal submersion with horizontal distribution. If $\sigma$ is a geodesic
of $\mathbf{M}$, then $\pi\circ\sigma$ is a geodesic of $\mathbf{B}$ if and
only if

$\displaystyle\pi_{*}(2A_{X}U+T_{U}U))+2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel
X\parallel^{2},$

where $X=\mathcal{H}(\sigma^{{}^{\prime}})$ and
$U=\mathcal{V}(\sigma^{{}^{\prime}})$ and $\parallel
X\parallel^{2}=g_{m}(X,X)$.

###### Proof

Since $\sigma$ is a geodesic on $\mathbf{M}$ from the equation (15)

$\displaystyle\sigma_{*}^{{}^{\prime\prime}}$ $\displaystyle=$
$\displaystyle\pi_{*}(grad_{\pi}\phi)\parallel
X\parallel^{2}-\pi_{*}(2A_{X}U+T_{U}U)-2d\phi(X)\pi_{*}X.$

Hence, $\pi\circ\sigma$ is a geodesic on $\mathbf{B}$ if and only if

$\displaystyle\pi_{*}(2A_{X}U+T_{U}U))+2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel
X\parallel^{2}.$

###### Remark 2

If $\sigma$ is a horizontal geodesic (that is, $\sigma$ is a geodesic with
$\mathcal{V}(\sigma^{{}^{\prime}})=0$), then $\pi\circ\sigma$ is a geodesic if
and only if $2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel
X\parallel^{2}$.

###### Definition 7

Let $\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ be
a conformal submersion with horizontal distribution and $\alpha$ be a smooth
curve in $\mathtt{B}$. Let $\alpha^{{}^{\prime}}$ be the tangent vector field
of $\alpha$ and $\tilde{(\alpha^{{}^{\prime}})}$ be its horizontal lift.
Define the horizontal lift of $\alpha$ as the integral curve $\sigma$ on
$\mathbf{M}$ of $(\tilde{\alpha^{{}^{\prime}})}$.

Now, we have

###### Proposition 2

Let $\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ be
a conformal submersion with horizontal distribution such that $A_{Z}Z=0$ for
all horizontal vector fields $Z$. Then, every horizontal lift of a geodesic of
$\mathbf{B}$ is a geodesic of $\mathbf{M}$ if and only if
$2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel X\parallel^{2}$, where $X$
is the horizontal part of the tangent vector field of the horizontal lift of
the geodesic on $\mathbf{B}$.

###### Proof

Let $\alpha$ be a geodesic on $\mathbf{B}$, $\sigma$ be the horizontal lift of
$\alpha$. Then, we have $\pi\circ\sigma=\alpha$ and $\sigma^{{}^{\prime}}(t)$
= $\tilde{(\alpha^{{}^{\prime}}(t))}$. Let
$X=\mathcal{H}(\sigma^{{}^{\prime}}(t))$ and
$U=\mathcal{V}(\sigma^{{}^{\prime}}(t))$, clearly
$X=\tilde{(\alpha^{{}^{\prime}}(t))}$ and $U=0$. Then, from the equations (15)
and (16)

$\displaystyle\pi_{*}(\mathcal{H}(\sigma^{{}^{\prime\prime}}))$
$\displaystyle=$
$\displaystyle\alpha^{{}^{\prime\prime}}-\pi_{*}(grad_{\pi}\phi)\parallel
X\parallel^{2}+2X(\phi)\pi_{*}X.$
$\displaystyle\mathcal{V}(\sigma^{{}^{\prime\prime}})$ $\displaystyle=$
$\displaystyle A_{X}X.$

Since $\alpha$ is a geodesic and $A_{X}X=0$ we have,
$\sigma^{{}^{\prime\prime}}=0$ if and only if
$2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel X\parallel^{2}$. That is,
every horizontal lift of a geodesic of $\mathbf{B}$ is a geodesic of
$\mathbf{M}$ if and only if $2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel
X\parallel^{2}$.

###### Corollary 3

Let $\pi:(\mathbf{M},\nabla,g_{m})\rightarrow(\mathbf{B},\nabla^{*},g_{b})$ be
a conformal submersion with horizontal distribution such that $A_{Z}Z=0$ for
all horizontal vector fields $Z$. Then, $\nabla^{*}$ is geodesically complete
if $\nabla$ is geodesically complete and
$2X(\phi)\pi_{*}X=\pi_{*}(grad_{\pi}\phi)\parallel X\parallel^{2}$, where $X$
is the horizontal part of the tangent vector field of the horizontal lift of
the geodesic on $\mathbf{B}$.

###### Proof

Let $\alpha$ be a geodesic of $\mathbf{B}$ and $\tilde{\alpha}$ be its
horizontal lift to $\mathbf{M}$, by Proposition (2) $\tilde{\alpha}$ is a
geodesic on $\mathbf{M}$. Since $\nabla$ is geodesically complete,
$\tilde{\alpha}$ can be defined on the entire real line. Then, the projected
curve of the extension of $\tilde{\alpha}$ is a geodesic and is the extension
of $\alpha$, that is $\nabla^{*}$ is geodesically complete.

## Acknowledgement

Mahesh T V thanks Indian Institute of Space Science and Technology (IIST),
Department of Space, Government of India for the award of Doctoral Research
Fellowship.

## References

  * (1) Abe, N., Hasegawa, K.: An affine submersion with horizontal distribution and its applications.  Differential Geometry and its Applications 14(3), 235–250 (2001)
  * (2) Fuglede, B.: Harmonic morphisms between Riemannian manifolds.  In: Annales de l’institut Fourier, vol. 28, pp. 107–144 (1978)
  * (3) Gudmundsson, S.: On the geometry of harmonic morphisms.  Mathematical Proceedings of the Cambridge Philosophical Society 108, 461–466 (1990)
  * (4) Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions.  J.Math Kyoto University 19, 215–229 (1979)
  * (5) O’Neill, B.: The fundamental equations of a submersion.  Michigan Mathematical Journal 13(4), 459–469 (1966)
  * (6) O’Neill, B.: Submersions and geodesics.  Duke Mathematical Journal 34(2), 363–373 (1967)
  * (7) Ornea, L., Romani, G.: The fundamental equations of conformal submersions.  Beitrage Algebra Geom 34(2), 233–243 (1993)


# On merit and equity in math

Alexander Givental

###### Abstract.

In recent paper Quantifying Inequities and Documenting Elitism in PhD-granting
Mathematical Sciences Departments in the United States [1] by a group of
accomplished and/or aspiring mathematicians, the authors use data to challenge
“the idea that the mathematical sciences in the United States is a
meritocracy”. We show that, regardless of the validity of the challenged idea,
the arguments in the paper are invalid. Namely, among other flaws, they rely
on a logical trick of asserting the validity of the conclusion derived from an
assumption which the authors neglect to test (and which actually contradicts
their own data).

My first refutation of paper [1] is contained in another article [2] (rejected
by the arXiv, apparently, for its not entirely academic character), which uses
a number of very elementary mathematical models to dispel several popular ways
of “social justice” reasoning. The models are so elementary that I wouldn’t
bother writing a separate article about them, but the aim of exposing the
fallacies of [1] turned out to be a sufficiently worthy occasion. The present
text is an abbreviated (and more “academic”) version of [2], focusing on the
latter task only.

The inequities, allegedly quantified in [1], concern: (a) underrepresentation
of women among math faculty, and (b) disproportional over-funding of
“elite”math departments by the NSF’s Department of Mathematical Sciences (DMS)
program. Below we expose a number of elementary mathematical, logical, and
factual errors committed in [1], which invalidate the methodology of the paper
and undermine its conclusions.

1. According to the authors of [1], in the past decade, women account for $30\%$ of all new PhD degrees awarded in math, and therefore (they argue) in pure meritocracy the portion of women on the math faculty of PhD-granting mathematics department had to be also $30\%$. The banal fallacy of this reasoning can be described by a preschool-level math theorem: the maximum is equal the average only when the quantity is constant. Namely, since the portion of women awarded PhD in math has been growing, even if women form $30\%$ of new hires, women still wouldn’t make 30% of all math faculty including those hired in the past. For instance, in my home department (see [3]) women currently make $13\%$ of all Senate faculty (which includes emeriti), $20\%$ of currently active, and $30\%$ among assistant and associate professors.

2. Even assuming (for the sake of the argument) that the quantity is constant, i.e. that women receive $30\%$ of PhD degrees in math for the span of an entire generation, the conclusion in the paper that “math in the US is not a meritocracy” would be based on explicitly formulated assumptions (see page 2 in [1]) which can be summarized this way: It can be expected that women form a certain fraction (e.g. $30\%$) of all people interested in pursuing a particular career (e.g. in math), but then in pure meritocracy they would constitute that same fraction in every tier of the achievement ladder (such as PhD degree or — a higher tier — professorship).

The obvious fallacy of conclusions derived from an assumption is that the
assumption might be false. And this particular assumption is well-known to be
generally false. Indeed, while it is possible that non-proportional attrition
rates of men and women up the achievement ladder are due to biases, there are
many examples where it would be hard to suspect any non-meritorious forces in
play. A famous one is chess [4] where among FIDE-rated players 11% are women
though among grandmasters they make $<3\%$. Another (random) example is found
in a study [5] of Germany’s several-tier system of selection of high-schoolers
to the International Chemical Olympiad, where the portion of girls decreased
with each tier (in 2017 from 48% to 0), and a similar decrease is found among
the recipients of awards (honorable mention, bronze, silver, gold) at the
Olympiads themselves.

Actually the very fact that women are being awarded $30\%$ of PhD degrees in
math (rather than $50\%$) already invalidates the authors’ assumption — unless
they also attribute this disproportion to bias against women (and hence bias
against men in other subjects, because [6] overall women in the US earn more
PhD degrees than men each year starting 2006).

We refer to [2] for the discussion of one illustrative model showing that the
hypothesis about the natural proportionality of the distributions of men and
women over all tiers of the achievement ladder has no grounds whatsoever. Even
under the assumption of equal standard deviations (true for distributions of
FIDE-rated players [4], and useful anyway since the opposite costed one
Harvard president their job [7]), the distributions may have different means
(as it happens, e.g with ACT math scores [8]). In the example of normal
distributions this leads to exponentially decaying ratio.

3. Concerning DMS funding of mathematics departments, the authors of [1] arrive (on page 9) at the conclusion that the funding “is not meritocratic and reinforces elitism”. They base this conclusion on their finding that the top $25\%$ of the departments ordered by their prestige ranking (whatever this means) absorb $65\%$ of DMS funds allocated to all departments so ranked, and that the top $20\%$ of all awardees (PIs? departments? — unclear) absorb $86\%$ of all DMS funds.

The obvious disconnect between the finding and conclusion manifests in the
absence of any attempt to take into account the scientific and educational
impact of those elite departments — or merely their size! What if those most
prestigious $25\%$ account for $65\%$ of all math faculty? Even if not, what
if they account for the overwhelming majority of math PhD students (to be
supported by the DMS funds)? What about their scientific impact (for
evaluation of which there are any number of metrics, based on citation
indexes, journal rankings, academy memberships, prizes and medals, etc. and —
yes — NSF awards)? The authors speculate along the lines of “the rich-get-
richer”, i.e. that better funding is not a result of success in math but a
primary cause of it. Yet they have absolutely no data supporting this fantasy.

4. Eventually, on page 10, in the spirit of 1984, they advocate for what amounts to imposing racial, ethnic, and gender quotas in hiring: “we advocate for the redefinition of ’prestige’ and ’elitism’ in mathematics in a way that better reflects equity for excellence. For example, institutions that might deserve our respect are institutions that reflect the diversity of the US population among their faculty and doctoral graduates in mathematics.”

To illustrate this new $raison\ d^{\prime}\hat{e}tre$ of mathematics
departments, they present the list of most prestigious 10: “The top ten PhD-
granting Mathematics departments by the percentage of women.” In fact, as it
is discussed with some detail in [2], while four of the ten can be estimated
to yield altogether about 8 math PhDs per year (for comparison: UC Berkeley
Math department awards 25-30), in the remaining six I didn’t manage to locate
any PhD programs in math at all.

The “non-academic” question somehow addressed in [2] is why this
scientifically looking yet grotesquely misleading paper was written at all. It
is inconceivable that a group of mathematicians could inadvertently make
errors as naive as some noted above. Alternatively, one could insist that this
is a typical case of cynical exploitation of the now-popular ideology for
personal gain. Yet, in [2], exercising the presumption of innocence, I argue
that the paper is a hoax, intended to compromise the DEI axiomatics, and to
illuminate the sheer incompatibility of meritocracy with equity.

## References

  * [1] R. Buckmire et al,  arXiv:2308.13750
  * [2] A. Givental,  On a “social justice” hoax
  * [3] UC Berkeley Math Senate Faculty
  * [4] Visualizing FIDE chess-rated list
  * [5] First steps toward gender equity in the chemistry Olympiad
  * [6] Number of doctoral degrees earned in the United States by gender
  * [7] Wikipedia article “Lawrence Summers”
  * [8] ACT score averages and standard deviations