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An almost complex structure on $M$ is equivalent to a reduction of the structure group of the tangent bundle

5

3

$begingroup$

Let $M$ be an $2n$-dimensional manifold. Let $mathcalF_mathrmGL(2n, mathbbR)$ be the frame bundle over $M$. Consider the subgroup $mathrmGL(n, mathbbC)subsetmathrmGL(2n, mathbbR)$. What I'm trying to prove is:

If $M$ has an almost complex structure $J:TMrightarrow TM$ then there is a reduction of the structure group $mathrmGL(2n, mathbbR)$ of $mathcalF_mathrmGL(2n, mathbbR)$ to $mathrmGL(n, mathbbC)$.

The reciprocal is also true, and I was able to prove it. But I'm stuck on this direction. Does anyone have a suggestion?

differential-geometry principal-bundles complex-manifolds almost-complex

share|cite|improve this question

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is your definition for reduction of the structure group?
  $endgroup$
  – Amitai Yuval
  Mar 25 at 7:12
  

  
  
  
  

add a comment | 

5

3

$begingroup$

Let $M$ be an $2n$-dimensional manifold. Let $mathcalF_mathrmGL(2n, mathbbR)$ be the frame bundle over $M$. Consider the subgroup $mathrmGL(n, mathbbC)subsetmathrmGL(2n, mathbbR)$. What I'm trying to prove is:

If $M$ has an almost complex structure $J:TMrightarrow TM$ then there is a reduction of the structure group $mathrmGL(2n, mathbbR)$ of $mathcalF_mathrmGL(2n, mathbbR)$ to $mathrmGL(n, mathbbC)$.

The reciprocal is also true, and I was able to prove it. But I'm stuck on this direction. Does anyone have a suggestion?

differential-geometry principal-bundles complex-manifolds almost-complex

share|cite|improve this question

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is your definition for reduction of the structure group?
  $endgroup$
  – Amitai Yuval
  Mar 25 at 7:12
  

  
  
  
  

add a comment | 

$begingroup$

Let $M$ be an $2n$-dimensional manifold. Let $mathcalF_mathrmGL(2n, mathbbR)$ be the frame bundle over $M$. Consider the subgroup $mathrmGL(n, mathbbC)subsetmathrmGL(2n, mathbbR)$. What I'm trying to prove is:

If $M$ has an almost complex structure $J:TMrightarrow TM$ then there is a reduction of the structure group $mathrmGL(2n, mathbbR)$ of $mathcalF_mathrmGL(2n, mathbbR)$ to $mathrmGL(n, mathbbC)$.

The reciprocal is also true, and I was able to prove it. But I'm stuck on this direction. Does anyone have a suggestion?

differential-geometry principal-bundles complex-manifolds almost-complex

share|cite|improve this question

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

$endgroup$

share|cite|improve this question

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

share|cite|improve this question

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

edited Mar 24 at 18:08

J. W. Tanner

4,7721420

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

asked Mar 24 at 17:49

Leonardo SchultzLeonardo Schultz

31116

1 Answer
1

active

oldest

votes

1

$begingroup$

Let us interpret the points in the frame bundle over $xin M$ as linear isomorphisms from $u:mathbb R^2nto T_xM$. Fixing an identification of $mathbb R^2n$ with $mathbb C^n$, you can view each $u$ as a real linear isomorphism $mathbb C^nto T_xM$. Now over $xin M$For a point $xin M$, consider the subset of those linear isomorphisms for which $u(iz)=J_x(u(z))$ for all $zinmathbb C^n$. It is clear that for such an isomorphism $u$ and $Ain GL(2n,mathbb R)$ (which here has to be viewed as the group of real linear isomorphisms from $mathbb C^n$ to itself, $ucirc A$ lies in the subset if and only if $A$ lies in the subgroup $GL(n,mathbb C)$. Taking a local smooth section $sigma$ of the frame bundle you can construct a section having values in the suspace in each fiber via $tildesigma(x)(z)=tfrac12(sigma(x)(z)-J_x(sigma(x)(iz))$. Thus we have defined a reduction of structure group.

share|cite|improve this answer

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923

$endgroup$

add a comment | 

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1

$begingroup$

Let us interpret the points in the frame bundle over $xin M$ as linear isomorphisms from $u:mathbb R^2nto T_xM$. Fixing an identification of $mathbb R^2n$ with $mathbb C^n$, you can view each $u$ as a real linear isomorphism $mathbb C^nto T_xM$. Now over $xin M$For a point $xin M$, consider the subset of those linear isomorphisms for which $u(iz)=J_x(u(z))$ for all $zinmathbb C^n$. It is clear that for such an isomorphism $u$ and $Ain GL(2n,mathbb R)$ (which here has to be viewed as the group of real linear isomorphisms from $mathbb C^n$ to itself, $ucirc A$ lies in the subset if and only if $A$ lies in the subgroup $GL(n,mathbb C)$. Taking a local smooth section $sigma$ of the frame bundle you can construct a section having values in the suspace in each fiber via $tildesigma(x)(z)=tfrac12(sigma(x)(z)-J_x(sigma(x)(iz))$. Thus we have defined a reduction of structure group.

share|cite|improve this answer

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923

$endgroup$

add a comment | 

1

$begingroup$

Let us interpret the points in the frame bundle over $xin M$ as linear isomorphisms from $u:mathbb R^2nto T_xM$. Fixing an identification of $mathbb R^2n$ with $mathbb C^n$, you can view each $u$ as a real linear isomorphism $mathbb C^nto T_xM$. Now over $xin M$For a point $xin M$, consider the subset of those linear isomorphisms for which $u(iz)=J_x(u(z))$ for all $zinmathbb C^n$. It is clear that for such an isomorphism $u$ and $Ain GL(2n,mathbb R)$ (which here has to be viewed as the group of real linear isomorphisms from $mathbb C^n$ to itself, $ucirc A$ lies in the subset if and only if $A$ lies in the subgroup $GL(n,mathbb C)$. Taking a local smooth section $sigma$ of the frame bundle you can construct a section having values in the suspace in each fiber via $tildesigma(x)(z)=tfrac12(sigma(x)(z)-J_x(sigma(x)(iz))$. Thus we have defined a reduction of structure group.

share|cite|improve this answer

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923

$endgroup$

add a comment | 

$begingroup$

Let us interpret the points in the frame bundle over $xin M$ as linear isomorphisms from $u:mathbb R^2nto T_xM$. Fixing an identification of $mathbb R^2n$ with $mathbb C^n$, you can view each $u$ as a real linear isomorphism $mathbb C^nto T_xM$. Now over $xin M$For a point $xin M$, consider the subset of those linear isomorphisms for which $u(iz)=J_x(u(z))$ for all $zinmathbb C^n$. It is clear that for such an isomorphism $u$ and $Ain GL(2n,mathbb R)$ (which here has to be viewed as the group of real linear isomorphisms from $mathbb C^n$ to itself, $ucirc A$ lies in the subset if and only if $A$ lies in the subgroup $GL(n,mathbb C)$. Taking a local smooth section $sigma$ of the frame bundle you can construct a section having values in the suspace in each fiber via $tildesigma(x)(z)=tfrac12(sigma(x)(z)-J_x(sigma(x)(iz))$. Thus we have defined a reduction of structure group.

share|cite|improve this answer

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923

$endgroup$

share|cite|improve this answer

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923

answered Mar 24 at 19:01

Andreas CapAndreas Cap

11.4k923