Nicholas Kayban, Department of Pure Mathematics, University of Waterloo
"Riemannian Submersions and the O'Neill Tensors"
In
an
introductory
Riemannian
geometry
course,
one
typically
encounters
the
Euler,
Gauss,
and
Codazzi
equations,
which
relate
the
curvature
of
a
submanifold
to
the
curvature
of
the
ambient
manifold
via
the
second
fundamental
form.
The
O'Neill
equations
are
analogous
equations
for
the
case
of
a
Riemannian
submersion.
In
this
talk
we
define
Riemannian
submersions
and
discuss
the
Fubini
Study
metric
on
$CP^n$
as
an
example.
We
also
consider
a
vector
bundle
$E$
over
a
Riemannian
manifold
$(M,g)$
where
the
$E$
is
endowed
with
a
Riemannian
metric
induced
from
a
fibre
metric
on
$E$,
a
connection
on
$E$,
and
the
Riemannian
metric
$g$
on
$M$,
such
that
the
projection
is
a
Riemannian
submersion.
The
O'Neill
tensors
are
defined,
and
we
state
the
fundamental
equations.
We
determine
the
O'Neill
tensors
of
the
Fubini-Study
metric
and
the
Riemannian
metric
on
$E$.
The
O'Neill
tensors
are
then
applied
to
show
that
the
sectional
curvature
of
the
Fubini-Study
metric
is
bounded
between
1
and
4.
Zoom meeting:
- Meeting ID: 958 7361 8652
- Passcode: 577854