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Pullback of 1-forms (complex 1,1-forms) with respect to a linear map
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Consider a linear map $f:N rightarrow M$ between two manifolds of dimension $n, m$. Let $alpha$ denote a 1-form on $M$. After choosing coordinates, we can assume that $alpha$ is the product of two $m times 1$ vectors: $alpha = v w^t$, where the entries of $v$ are smooth functions and the entries of $w$ are the basis elements which we can name $dx_1, ldots, dx_m$. Let $A$ denote the matrix corresponding to the morphism $f$.
I am interested in the pullback of $alpha$ with respect to $f$. How can it be written in terms of matrix-vector multiplication?
Also, in the case where $M$ and $N$ are Kähler manifolds, a (1,1) form is represented by a matrix. Again, can the pullback be described in terms of matrix multiplication?
differential-geometry multilinear-algebra pullback
asked Jul 19 at 18:21
arla
183
add a comment |Â
up vote
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Consider a linear map $f:N rightarrow M$ between two manifolds of dimension $n, m$. Let $alpha$ denote a 1-form on $M$. After choosing coordinates, we can assume that $alpha$ is the product of two $m times 1$ vectors: $alpha = v w^t$, where the entries of $v$ are smooth functions and the entries of $w$ are the basis elements which we can name $dx_1, ldots, dx_m$. Let $A$ denote the matrix corresponding to the morphism $f$.
I am interested in the pullback of $alpha$ with respect to $f$. How can it be written in terms of matrix-vector multiplication?
Also, in the case where $M$ and $N$ are Kähler manifolds, a (1,1) form is represented by a matrix. Again, can the pullback be described in terms of matrix multiplication?
differential-geometry multilinear-algebra pullback
asked Jul 19 at 18:21
arla
183
3
You need to have a linear structure on your manifolds in order to talk about linear maps on them. What is that structure?
– Jackozee Hakkiuz
Jul 19 at 18:29
I'm just assuming that they are real/complex spaces, so $mathbbR^n$ or $mathbbC^n$
– arla
Jul 20 at 8:23
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider a linear map $f:N rightarrow M$ between two manifolds of dimension $n, m$. Let $alpha$ denote a 1-form on $M$. After choosing coordinates, we can assume that $alpha$ is the product of two $m times 1$ vectors: $alpha = v w^t$, where the entries of $v$ are smooth functions and the entries of $w$ are the basis elements which we can name $dx_1, ldots, dx_m$. Let $A$ denote the matrix corresponding to the morphism $f$.
I am interested in the pullback of $alpha$ with respect to $f$. How can it be written in terms of matrix-vector multiplication?
Also, in the case where $M$ and $N$ are Kähler manifolds, a (1,1) form is represented by a matrix. Again, can the pullback be described in terms of matrix multiplication?
differential-geometry multilinear-algebra pullback
asked Jul 19 at 18:21
arla
183
Consider a linear map $f:N rightarrow M$ between two manifolds of dimension $n, m$. Let $alpha$ denote a 1-form on $M$. After choosing coordinates, we can assume that $alpha$ is the product of two $m times 1$ vectors: $alpha = v w^t$, where the entries of $v$ are smooth functions and the entries of $w$ are the basis elements which we can name $dx_1, ldots, dx_m$. Let $A$ denote the matrix corresponding to the morphism $f$.
I am interested in the pullback of $alpha$ with respect to $f$. How can it be written in terms of matrix-vector multiplication?
Also, in the case where $M$ and $N$ are Kähler manifolds, a (1,1) form is represented by a matrix. Again, can the pullback be described in terms of matrix multiplication?
differential-geometry multilinear-algebra pullback
asked Jul 19 at 18:21
arla
183
asked Jul 19 at 18:21
arla
183
asked Jul 19 at 18:21
arla
183
asked Jul 19 at 18:21
arla
183
183
3
You need to have a linear structure on your manifolds in order to talk about linear maps on them. What is that structure?
– Jackozee Hakkiuz
Jul 19 at 18:29
I'm just assuming that they are real/complex spaces, so $mathbbR^n$ or $mathbbC^n$
– arla
Jul 20 at 8:23
add a comment |Â
3
You need to have a linear structure on your manifolds in order to talk about linear maps on them. What is that structure?
– Jackozee Hakkiuz
Jul 19 at 18:29
I'm just assuming that they are real/complex spaces, so $mathbbR^n$ or $mathbbC^n$
– arla
Jul 20 at 8:23
3
3
You need to have a linear structure on your manifolds in order to talk about linear maps on them. What is that structure?
– Jackozee Hakkiuz
Jul 19 at 18:29
You need to have a linear structure on your manifolds in order to talk about linear maps on them. What is that structure?
– Jackozee Hakkiuz
Jul 19 at 18:29
I'm just assuming that they are real/complex spaces, so $mathbbR^n$ or $mathbbC^n$
– arla
Jul 20 at 8:23
I'm just assuming that they are real/complex spaces, so $mathbbR^n$ or $mathbbC^n$
– arla
Jul 20 at 8:23
add a comment |Â
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3
You need to have a linear structure on your manifolds in order to talk about linear maps on them. What is that structure?
– Jackozee Hakkiuz
Jul 19 at 18:29
I'm just assuming that they are real/complex spaces, so $mathbbR^n$ or $mathbbC^n$
– arla
Jul 20 at 8:23