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Hello. I completed my PhD in differential geometry in Edinburgh. I have been reading MathOverflow for a year and a half two and a half years three and a half years.

My primary interests are in $G_2$ and $Spin_7$ geometry.


Nov
17
comment Isometries of hyper-Kähler manifolds
@BenMcKay, I see the confusion coming from the notation $SO(Im \mathbb{H})$. By this I mean the group of rotations of the $3$-dim real vector space $Im \mathbb{H}$ spanned by the parallel complex structures, not something whose entries lie in $Im \mathbb{H}$.
Nov
17
revised Isometries of hyper-Kähler manifolds
Corrected a statement pointed out by a comment, added images and corrected the classification of rotation subgroups.
Nov
17
comment Isometries of hyper-Kähler manifolds
@YangMills, I ought to be more careful with my well-known 'facts'. I'll fix it now.
Nov
9
revised Isometries of hyper-Kähler manifolds
deleted 58 characters in body
Nov
9
comment Isometries of hyper-Kähler manifolds
@RobertBryant, yes of course, I should have noticed that. I have edited the question to clarify.
Nov
9
revised Isometries of hyper-Kähler manifolds
added 203 characters in body
Nov
8
awarded  Student
Nov
8
asked Isometries of hyper-Kähler manifolds
Oct
29
comment Dropping the closed requirement from the symplectic manifold definition?
@OldřichSpáčil, I believe it's in Gray and Hervella's The sixteen classes of almost Hermitian manifolds and their linear invariants (1980), but I can't seem to find the paper in my collection (and I don't belong to an institution). You get a finer classification when you have the full almost-Hermitian structure, which is the main part of the paper. The idea is to decompose $\Lambda^3V$ into irreducible reps of the non-compact symplectic group, and apply that to $d\omega$. Similar classifications exist for various other structures, and there are subsequent papers.
Oct
28
comment Dropping the closed requirement from the symplectic manifold definition?
While it's true that the condition of admitting such a reduction is topology, once you have one the situation is more geometrical: when $\omega$ is closed there is no local structure but now there is, namely $d\omega$ itself. These geometries can be classified into four types.
Oct
26
comment Trivial representation in tensor square
Frank Adams' book Lectures on Lie Groups explains these things very clearly.
Oct
13
comment Manifolds admitting flat connections
Related to Q1: mathoverflow.net/questions/91852/…
Oct
12
comment Does local reducibility imply global reducibility of universal covering?
You at least need completeness. Otherwise just take any non-rectangular simply connected region in the plane.
Sep
24
awarded  Autobiographer
May
21
comment What is an element of an iterated tangent bundle?
@Ryan, e.g. the second order tangent bundle of $\mathcal{M}$ is the vector bundle over $\mathcal{M}$ whose fibre at $p$ consists of $2$-jets at $p$ of curves through $p$. It has rank $n(n+1)/2$ for $n = dim \mathcal{M}$. Just as for the first order tangent bundle, there is also an algebraic definition if you are so minded.
May
21
comment What is an element of an iterated tangent bundle?
Whatever name you choose, it shouldn't allow confusion with elements of higher-order tangent bundles (as opposed to iterated ones).
Apr
26
comment Relation between kahler potential and Hermitian metric
If $h$ is the Hermitian form, what is the meaning of $\log h$?
Jan
4
comment Metalinear frame bundle on sphere or $\mathbb{C}P^n$
What do you want the complex vector bundle $P$ to be for your three spaces? And, given such a $P$, is there not often a choice of such structure?
Jan
3
comment Holonomy group of a fiber bundle
I agree with Igor; $M = \{\bullet\}$, $N$ any flat space with non-trivial holonomy and $G$ trivial gives a counterexample.
Jan
3
comment Holonomy group of a fiber bundle
Isn't it obvious when using restricted holonomy?