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FRIDAY, OCTOBER 31, 2014
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Andy Manion
A new method of constructing connections on vector bundles
Liviu Nicolaescu
This post was suggested by a question on the MathOverflow site. After I answered part of it I noticed that it is related to a recent work of mine of a probabilistic nature. What follows involves no probability. As far as the terminology concerning connections, I'll stick to the terminology in Section 3.3. of my book.
obico
There was an error in this gadge
Suppose that M is a smooth manifold of dimension E → M is a real, smooth vector bundle of rank ν over M . We define a pairing on E to be a section of the bundle E ∗ ⊠ E ∗ → M × M, where E ∗ ⊠ E ∗ is the vector bundle π1∗ E ∗ ⊗ π2∗ E ∗ , πi (x1 , x2 ) = xi, ∀(x1 , x2 ) ∈ M × M, i = 1, 2. For x, y
∈ M
we can view Bx,y
∗
Gowers's Weblog
as a bilinear map
∗
∈ Ex × Ey
MY BLOG LIST
I’m not getting the feeling tha problem is taking off as a Pol myself like the problem enoug 4 days ago
Bx,y : E x × E y → R.
What's new
Suppose one has a bounded numbers. What kinds of limits sequence? Of course, we hav , wh... 4 days ago
This induces a linear map ∗
S x,y = S (B)x,y : E y → E x .
We say that the pairing is nondegenerate if for any x isomorphism
∈ M
the bilinear map B(x, x)
is nondegenerate. In particular, this induces an
: Ex × Ex → R
∗
1
BLOG ARCHIVE
S x = S x,x : E x → E x .
► 2017 (3) ► 2016 (5)
We obtain tunneling operators
► 2015 (12) ▼ 2014 (16) −1
T (x, y) = S x
► November (3)
S x,y : E y → E x .
▼ October (2)
A new method of construc
Fix an open coordinate patch O ⊂ M with coordinates (xi )1≤i≤m . Assume O is sufficiently small so E trivializes over O . Suppose that ν e (x) = (e α (x))1≤α≤ν is a local frame of E over O . We denote by R the trivial vector bundle R × O → O. O
− −
− −
The local frame e defines a bundle isomorphism Φ( e ) − −
− −
: R → EO − −O
. In the local frame e the tunelling are represented by a tunneling map − −
ν
T e : O × O → End(R ), − −
The Unreasonable Effectiv ► August (3) ► May (1) ► April (2)
−1
T e (x, y) = Φx ( e ) − − − −
► March (2) T (x, y)Φy ( e ). − −
► January (3) ► 2013 (23)
Note that T− e (x, x) −
= 1Rν
. For i
► 2012 (47)
define
= 1, … , m
ν
Γi ( e ) : O → End(R ), − −
Γi ( e , x) = −∂xi T e (x, y)∣ . ∣ − − y=x − −
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We set
Posts Comments
m i
Γ( e , x) = ∑ Γi ( e , x)dx − − − −
= −dx T (x, y)∣ ∣
ν
y=x
∈ End( R
1
) ⊗ Ω (O ),
i=1
TOTAL PAGEVIEWS
where dx denotes the differential (exterior derivative) with respect to the xvariables. If f is another local frame of EO , then there exists a smooth map ν
− −
such that
g : O → Aut(R )
Φx ( f ) = Φx ( e ) ∘ g(x), − − − −
http://liviusmathblog.blogspot.com/2014/10/a-new-method-of-constructing.html
22,776
∀x ∈ O .
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Then −1
T f (x, y) = g(x)
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T e (x, y)g(y), − −
− −
Email address...Submit −1
Γ( f , x) = −dx ( g(x) − −
This proves that the correspondence e ↦ − − the nondegenerate pairing B .
)∣ ∣
y=x
T e (x, x)g(yx + g
−1
− −
−1
(x)Γ( e , x)g(x) = g(x) − −
dg(x) + g
−1
(x)Γ( e , x)g(x). − −
defines a connection on E . We will denote it by ∇B and we will refer to it as the connection associated to
Γ( e ) − −
Let us compute its curvature RB . Using the local frame e we can write − −
R
B
=
i
∑
Rij ( e , x)dx − −
j
ν
∧ dx
2
∈ End(R ) ⊗ Ω (O ),
1≤i
