Mixing
induced Levi-Civita connection on exterior power
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Let $M^n$ be a Riemannian manifold and $nabla$ the Levi-Civita connection of $M$. Let
beginalign*
Lambda^2(M) := coprod_p in M Lambda^2(T_p M).
endalign*
My question is, how one can define an induced connection
beginalign*
nabla: mathcalT(M) times Gamma(Lambda^2(M)) to Gamma(Lambda^2(M))
endalign* on the vector bundle $pi: Lambda^2(M) to M$, where
$Gamma(Lambda^2(M))$ is the set of smooth sections of $Lambda^2(M)$.
I.e. I want to understand the term
beginalign*
nabla_X (Y wedge Z)
endalign*
for $X,Y,Z in mathcalT(M)$ (and therefore $Y wedge Z in Gamma(Lambda^2(M)))$.
I would also appreciate any kind of reference where this is explained.
connections
asked Jul 17 at 10:08
user37662
11
add a comment |Â
up vote
0
down vote
favorite
Let $M^n$ be a Riemannian manifold and $nabla$ the Levi-Civita connection of $M$. Let
beginalign*
Lambda^2(M) := coprod_p in M Lambda^2(T_p M).
endalign*
My question is, how one can define an induced connection
beginalign*
nabla: mathcalT(M) times Gamma(Lambda^2(M)) to Gamma(Lambda^2(M))
endalign* on the vector bundle $pi: Lambda^2(M) to M$, where
$Gamma(Lambda^2(M))$ is the set of smooth sections of $Lambda^2(M)$.
I.e. I want to understand the term
beginalign*
nabla_X (Y wedge Z)
endalign*
for $X,Y,Z in mathcalT(M)$ (and therefore $Y wedge Z in Gamma(Lambda^2(M)))$.
I would also appreciate any kind of reference where this is explained.
connections
asked Jul 17 at 10:08
user37662
11
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $M^n$ be a Riemannian manifold and $nabla$ the Levi-Civita connection of $M$. Let
beginalign*
Lambda^2(M) := coprod_p in M Lambda^2(T_p M).
endalign*
My question is, how one can define an induced connection
beginalign*
nabla: mathcalT(M) times Gamma(Lambda^2(M)) to Gamma(Lambda^2(M))
endalign* on the vector bundle $pi: Lambda^2(M) to M$, where
$Gamma(Lambda^2(M))$ is the set of smooth sections of $Lambda^2(M)$.
I.e. I want to understand the term
beginalign*
nabla_X (Y wedge Z)
endalign*
for $X,Y,Z in mathcalT(M)$ (and therefore $Y wedge Z in Gamma(Lambda^2(M)))$.
I would also appreciate any kind of reference where this is explained.
connections
asked Jul 17 at 10:08
user37662
11
Let $M^n$ be a Riemannian manifold and $nabla$ the Levi-Civita connection of $M$. Let
beginalign*
Lambda^2(M) := coprod_p in M Lambda^2(T_p M).
endalign*
My question is, how one can define an induced connection
beginalign*
nabla: mathcalT(M) times Gamma(Lambda^2(M)) to Gamma(Lambda^2(M))
endalign* on the vector bundle $pi: Lambda^2(M) to M$, where
$Gamma(Lambda^2(M))$ is the set of smooth sections of $Lambda^2(M)$.
I.e. I want to understand the term
beginalign*
nabla_X (Y wedge Z)
endalign*
for $X,Y,Z in mathcalT(M)$ (and therefore $Y wedge Z in Gamma(Lambda^2(M)))$.
I would also appreciate any kind of reference where this is explained.
connections
asked Jul 17 at 10:08
user37662
11
asked Jul 17 at 10:08
user37662
11
asked Jul 17 at 10:08
user37662
11
asked Jul 17 at 10:08
user37662
11
11
add a comment |Â
add a comment |Â
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