Orientation on normal bundle
$begingroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2552925
$endgroup$
add a comment |
$begingroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2552925
$endgroup$
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
add a comment |
$begingroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2552925
$endgroup$
Let $M subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
differential-geometry algebraic-topology differential-topology vector-bundles
differential-geometry algebraic-topology differential-topology vector-bundles
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2552925
edited Jan 2 at 10:52
Bernard
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2552925
edited Jan 2 at 10:52
Bernard
122k740116
edited Jan 2 at 10:52
Bernard
122k740116
edited Jan 2 at 10:52
Bernard
122k740116
122k740116
asked Jan 2 at 10:48
CL.CL.
2,2552925
asked Jan 2 at 10:48
CL.CL.
2,2552925
asked Jan 2 at 10:48
CL.CL.
2,2552925
2,2552925
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
add a comment |
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01
add a comment |
1 Answer
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$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
add a comment |
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$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
add a comment |
$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
add a comment |
$begingroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
$endgroup$
The obstruction of the orientability is the first Stiefel Whitney class $w_1$. Suppose that $M$ is a submanifold of $N$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$ and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TNoplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
answered Jan 2 at 11:53
Tsemo AristideTsemo Aristide
58.8k11445
58.8k11445
add a comment |
add a comment |
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$begingroup$
This follows from the comments at the bottom of page 78 and top of page 79.
$endgroup$
– Tyrone
Jan 3 at 11:01