Meaning of commutativity of a local flow of a vector field and left-translations
In DoCarmo’s Riemannian Geometry book, it is writen that if $x_t$ is the flow of a left invariant vector field $X$ of a lie group $G$, then $L_ycirc x_t=x_t circ L_y$. But the domains of the left and right side do not necessarily coincide since the domain of $L_y$ is the whole $G$ while the domain of $x_t$ is in general a neighborhood of $G$, so does the author simply mean that for all $u$ in the domain of $x_t$ we have $L_ycirc x_t(u)=x_t(u)$, or what? That expression makes sense only if the domain of $x_t$ is also the whole $G$, but is it really so?
differential-geometry riemannian-geometry smooth-manifolds vector-fields
asked Nov 21 '18 at 12:23
User12239
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In DoCarmo’s Riemannian Geometry book, it is writen that if $x_t$ is the flow of a left invariant vector field $X$ of a lie group $G$, then $L_ycirc x_t=x_t circ L_y$. But the domains of the left and right side do not necessarily coincide since the domain of $L_y$ is the whole $G$ while the domain of $x_t$ is in general a neighborhood of $G$, so does the author simply mean that for all $u$ in the domain of $x_t$ we have $L_ycirc x_t(u)=x_t(u)$, or what? That expression makes sense only if the domain of $x_t$ is also the whole $G$, but is it really so?
differential-geometry riemannian-geometry smooth-manifolds vector-fields
asked Nov 21 '18 at 12:23
User12239
412216
add a comment |
In DoCarmo’s Riemannian Geometry book, it is writen that if $x_t$ is the flow of a left invariant vector field $X$ of a lie group $G$, then $L_ycirc x_t=x_t circ L_y$. But the domains of the left and right side do not necessarily coincide since the domain of $L_y$ is the whole $G$ while the domain of $x_t$ is in general a neighborhood of $G$, so does the author simply mean that for all $u$ in the domain of $x_t$ we have $L_ycirc x_t(u)=x_t(u)$, or what? That expression makes sense only if the domain of $x_t$ is also the whole $G$, but is it really so?
differential-geometry riemannian-geometry smooth-manifolds vector-fields
asked Nov 21 '18 at 12:23
User12239
412216
In DoCarmo’s Riemannian Geometry book, it is writen that if $x_t$ is the flow of a left invariant vector field $X$ of a lie group $G$, then $L_ycirc x_t=x_t circ L_y$. But the domains of the left and right side do not necessarily coincide since the domain of $L_y$ is the whole $G$ while the domain of $x_t$ is in general a neighborhood of $G$, so does the author simply mean that for all $u$ in the domain of $x_t$ we have $L_ycirc x_t(u)=x_t(u)$, or what? That expression makes sense only if the domain of $x_t$ is also the whole $G$, but is it really so?
differential-geometry riemannian-geometry smooth-manifolds vector-fields
differential-geometry riemannian-geometry smooth-manifolds vector-fields
asked Nov 21 '18 at 12:23
User12239
412216
asked Nov 21 '18 at 12:23
User12239
412216
edited Nov 21 '18 at 12:49
asked Nov 21 '18 at 12:23
User12239
412216
asked Nov 21 '18 at 12:23
User12239
412216
asked Nov 21 '18 at 12:23
User12239
412216
412216
add a comment |
add a comment |
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