**
Amanda
Petcu,
Department
of
Pure
Mathematics,
University
of
Waterloo**

**
"Some
Calculations
Regarding
$G_2$
and
the
Isometric
Flow
I"**

In
the
paper
[1]
the
authors
consider
the
following
setup
for
a
$7$-dimensional
manifold
$M$.
Given
a
hypersymplectic
structure
$\underline{\omega}$
on
$X^4$
they
consider
the
manifold
$M
=
X^4
\times
T^3$
where
$T^3
=
S^1
\times
S^1
\times
S^1$.
With
this
setup,
they
consider
a
closed
$3$-form
$\varphi$
on
$M$
that
gives
a
$G_2$
structure.
This
is
due
to
the
hypersymplectic
structure
on
$X^4$.
In
the
case
where
$X^4$
is
compact
the
authors
deform
the
hypersymplectic
structure
on
$X^4$
to
a
hyperkahler
triple.
Then
the
$G_2$-structure
$\varphi$
on
$M
=
X^4
\times
T^3
$
has
vanishing
torsion
forms
when
$\underline{\omega}$
is
a
hyperkahler
triple
implying
that
$\varphi$
determines
a
$G_2$-metric.

In
this
talk,
I
will
loosen
the
conditions
on
$X^4$
to
pre-hypersymplectic
and
compute
the
forms
$\varphi$
and
$*_{7}
\varphi
=
\psi$.
We
will
also
compute
the
four
torsion
forms
for
$M$
and
determine
what
conditions
are
needed
in
order
for
the
torsion
forms
to
vanish.
Finally
given
the
full
torsion
tensor
$T$
of
$M$,
we
will
compute
$\operatorname{Div}
T$
in
terms
of
the
four
torsion
forms
and
determine
what
conditions
we
might
need
in
order
for
$\operatorname{Div}
T$
to
vanish.

[1]
J.Fine
and
C.
Yao,
``Hypersymplectic
4-manifolds,
the
$G_2$-Laplacian
flow,
and
extension
assuming
bounded
scalar
curvature'',
Duke
University
Press,
vol.
167,
no.
18,
2018,
doi:
10.1215/00127094-2018-0040

MC 5417

Note special time for this seminar.