Mixing
$mathcal{C}^1$-topology of a submanifold with boundary
$begingroup$
Let $M subset mathbb{R}^n$ be a compact connected manifold without boundary embedded in $mathbb{R}^n$, then we can define the $mathcal{C}^1$-topology of the functions $mathcal{F}(M)= {f: Mto mathbb{R};$ $f$ is a smooth function$}$, as the topology induced by the sets
$$B(f,varepsilon) = { g: M to mathbb{R}; |f(x) -g(x)|<varepsilon, |text{d}f_x - text{d}g_x |< varepsilon, forall x in M}. $$
where $$|text{d}f_x - text{d}g_x | = sup_{v in T_xMsetminus{0}} left| (text{d}f_x -text{d}g_x)left(frac{v}{|v|}right) right|.$$
Let $A subset M$ be a compact connected submanifold with boundary and same dimension of $M$.
My Doubt Is the $mathcal{C}^1$- topology of $mathcal{F}(A)$ equal the
quotient topology induced by the function
begin{align*}phi:mathcal{F}(M) &to mathcal{F}(A) \
f &mapsto fBigvert_{A}quad quad?
end{align*}
differential-topology smooth-manifolds compact-manifolds
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
$endgroup$
add a comment |
$begingroup$
Let $M subset mathbb{R}^n$ be a compact connected manifold without boundary embedded in $mathbb{R}^n$, then we can define the $mathcal{C}^1$-topology of the functions $mathcal{F}(M)= {f: Mto mathbb{R};$ $f$ is a smooth function$}$, as the topology induced by the sets
$$B(f,varepsilon) = { g: M to mathbb{R}; |f(x) -g(x)|<varepsilon, |text{d}f_x - text{d}g_x |< varepsilon, forall x in M}. $$
where $$|text{d}f_x - text{d}g_x | = sup_{v in T_xMsetminus{0}} left| (text{d}f_x -text{d}g_x)left(frac{v}{|v|}right) right|.$$
Let $A subset M$ be a compact connected submanifold with boundary and same dimension of $M$.
My Doubt Is the $mathcal{C}^1$- topology of $mathcal{F}(A)$ equal the
quotient topology induced by the function
begin{align*}phi:mathcal{F}(M) &to mathcal{F}(A) \
f &mapsto fBigvert_{A}quad quad?
end{align*}
differential-topology smooth-manifolds compact-manifolds
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
$endgroup$
add a comment |
$begingroup$
Let $M subset mathbb{R}^n$ be a compact connected manifold without boundary embedded in $mathbb{R}^n$, then we can define the $mathcal{C}^1$-topology of the functions $mathcal{F}(M)= {f: Mto mathbb{R};$ $f$ is a smooth function$}$, as the topology induced by the sets
$$B(f,varepsilon) = { g: M to mathbb{R}; |f(x) -g(x)|<varepsilon, |text{d}f_x - text{d}g_x |< varepsilon, forall x in M}. $$
where $$|text{d}f_x - text{d}g_x | = sup_{v in T_xMsetminus{0}} left| (text{d}f_x -text{d}g_x)left(frac{v}{|v|}right) right|.$$
Let $A subset M$ be a compact connected submanifold with boundary and same dimension of $M$.
My Doubt Is the $mathcal{C}^1$- topology of $mathcal{F}(A)$ equal the
quotient topology induced by the function
begin{align*}phi:mathcal{F}(M) &to mathcal{F}(A) \
f &mapsto fBigvert_{A}quad quad?
end{align*}
differential-topology smooth-manifolds compact-manifolds
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
$endgroup$
Let $M subset mathbb{R}^n$ be a compact connected manifold without boundary embedded in $mathbb{R}^n$, then we can define the $mathcal{C}^1$-topology of the functions $mathcal{F}(M)= {f: Mto mathbb{R};$ $f$ is a smooth function$}$, as the topology induced by the sets
$$B(f,varepsilon) = { g: M to mathbb{R}; |f(x) -g(x)|<varepsilon, |text{d}f_x - text{d}g_x |< varepsilon, forall x in M}. $$
where $$|text{d}f_x - text{d}g_x | = sup_{v in T_xMsetminus{0}} left| (text{d}f_x -text{d}g_x)left(frac{v}{|v|}right) right|.$$
Let $A subset M$ be a compact connected submanifold with boundary and same dimension of $M$.
My Doubt Is the $mathcal{C}^1$- topology of $mathcal{F}(A)$ equal the
quotient topology induced by the function
begin{align*}phi:mathcal{F}(M) &to mathcal{F}(A) \
f &mapsto fBigvert_{A}quad quad?
end{align*}
differential-topology smooth-manifolds compact-manifolds
differential-topology smooth-manifolds compact-manifolds
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
edited Jan 25 at 14:28
Matheus Manzatto
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
asked Sep 9 '18 at 17:15
Matheus ManzattoMatheus Manzatto
1,3771625
1,3771625
add a comment |
add a comment |
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