Kosinski in his book "Differential Manifolds" states:

"A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc subbundle of a tubular neighbourhood of $M$".

He proves this statement like that:

"At first we reparametrize interior of $E$ to make it a vector bundle and then consider the unit disc subbundle $E'$. $E'$ can be expanded by an isotopy to cover $E$ and since it is compact this isotopy can be extended to an isotopy of $N$. The resulting isotopy will expand the interior of $E$ to a tubular neighbourhood of $M$ containing $E$ as a closed disc subbundle."

So, my question is: why this reparametrization is always possible?

P. S.: Kosinski defines a closed tubular neighbourhood as a subset of $N$, which has a stucture of a disc bundle over $M$ with $M$ as a zero section. Without words about structure group. There must be some kind of restrictions on it?

Kosinski in his book "Differential Manifolds" states:

"A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc subbundle of a tubular neighbourhood of $M$".

He proves this statement like that:

"At first we reparametrize interior of $E$ to make it a vector bundle and then consider the unit disc subbundle $E'$. $E'$ can be expanded by an isotopy to cover $E$ and since it is compact this isotopy can be extended to an isotopy of $N$. The resulting isotopy will expand the interior of $E$ to a tubular neighbourhood of $M$ containing $E$ as a closed disc subbundle."

So, my question is: why is this reparametrization always possible?

P. S.: Kosinski defines a closed tubular neighbourhood as a subset of $N$, that has a stucture of a disc bundle over $M$ with $M$ as a zero section. Without any words about structure group. Must there be some kind of restrictions on it?

Kosinski in his book "Differential Manifolds" states:

"A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc subbundle of a tubular neighbourhood of $M$".

He proves this statement like that:

"At first we reparametrize interior of $E$ to make it a vector bundle and then consider the unit disc subbundle $E'$. $E'$ can be expanded by an isotopy to cover $E$ and since it is compact this isotopy can be extended to an isotopy of $N$. The resulting isotopy will expand the interior of $E$ to a tubular neighbourhood of $M$ containing $E$ as a closed disc subbundle."

So, my question is: why is this reparametrization always possible?

P. S.: Kosinski defines a closed tubular neighbourhood as a subset of $N$, which has a stucture of a disc bundle over $M$ with $M$ as a zero section. Without words about structure group. There must be some kind of restrictions on it?

Kosinski in his book "Differential Manifolds" states:

"A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc subbundle of a tubular neighbourhood of $M$".

He proves this statement like that:

"At first we reparametrize interior of $E$ to make it a vector bundle and then consider the unit disc subbundle $E'$. $E'$ can be expanded by an isotopy to cover $E$ and since it is compact this isotopy can be extended to an isotopy of $N$. The resulting isotopy will expand the interior of $E$ to a tubular neighbourhood of $M$ containing $E$ as a closed disc subbundle."

So, my question is: why this reparametrization is always possible?

P. S.: Kosinski defines a closed tubular neighbourhood as a subset of $N$, which has a stucture of a disc bundle over $M$ with $M$ as a zero section. Without words about structure group. There must be some kind of restrictions on it?