What is the term for a “flat”, continuous set of points?
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What is the term for an $n$-dimensional manifold embedded in Euclidean $n+1$-space for which the function $textbf{p}(t)$ describing the geodesic (parameterized in terms of $t$) between two points has constant partial derivatives ($Vertfrac{d textbf{p}}{dt}Vert=k$ for some real or complex number $k$)? (e.g. a line, plane, etc.)
Alternatively, what is the term for continuous set $S$ of points in Euclidean $n$-space such that there exists a path $P$ between any two points in $S$ described by the funtion $textbf{f}:tto P$, where $Vertfrac{dtextbf{f}}{dt}Vert$ is constant?
Alternatively, alternatively, what is the term for an $n$-dimensional manifold $S$ embedded in Euclidean $n+1$-space such that the geodesic curvature is $0$ at all points in $S$?
i.e. what do you call an $n$-dimensional "flat thing"?
general-topology geometry differential-geometry
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
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add a comment |
$begingroup$
What is the term for an $n$-dimensional manifold embedded in Euclidean $n+1$-space for which the function $textbf{p}(t)$ describing the geodesic (parameterized in terms of $t$) between two points has constant partial derivatives ($Vertfrac{d textbf{p}}{dt}Vert=k$ for some real or complex number $k$)? (e.g. a line, plane, etc.)
Alternatively, what is the term for continuous set $S$ of points in Euclidean $n$-space such that there exists a path $P$ between any two points in $S$ described by the funtion $textbf{f}:tto P$, where $Vertfrac{dtextbf{f}}{dt}Vert$ is constant?
Alternatively, alternatively, what is the term for an $n$-dimensional manifold $S$ embedded in Euclidean $n+1$-space such that the geodesic curvature is $0$ at all points in $S$?
i.e. what do you call an $n$-dimensional "flat thing"?
general-topology geometry differential-geometry
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
$endgroup$
1
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Curves in the plane always have positive curvature; it's the radius of a certain tangent circle. You get negative curvatures in three dimensions.
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– MJD
Dec 17 '18 at 4:39
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A geodesic has constant speed by assumption, so $lVertfrac{mathrm{d}mathbf{p}}{mathrm{d}t}rVert=k$ is trivially true.
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– user10354138
Dec 17 '18 at 14:44
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$textbf{p}(t)$ is the parameterization of the geodesic in Euclidean $n+1$-space, i.e. $textbf{p}(t)=left(x_1(t),x_2(t),ldots,x_n(t)right)$ where $x_1,x_2,ldots,x_n$ are are orthogonal. I'm new to differential-geometry so I might not be saying that right, though.
$endgroup$
– R. Burton
Dec 17 '18 at 14:50
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Are you looking for Hyperplane? If not, how does the thing you describe differ from a hyperplane?
$endgroup$
– MvG
Dec 18 '18 at 0:01
$begingroup$
Yes, that is the thing I was talking about. THANK YOU!
$endgroup$
– R. Burton
Dec 18 '18 at 0:16
add a comment |
$begingroup$
What is the term for an $n$-dimensional manifold embedded in Euclidean $n+1$-space for which the function $textbf{p}(t)$ describing the geodesic (parameterized in terms of $t$) between two points has constant partial derivatives ($Vertfrac{d textbf{p}}{dt}Vert=k$ for some real or complex number $k$)? (e.g. a line, plane, etc.)
Alternatively, what is the term for continuous set $S$ of points in Euclidean $n$-space such that there exists a path $P$ between any two points in $S$ described by the funtion $textbf{f}:tto P$, where $Vertfrac{dtextbf{f}}{dt}Vert$ is constant?
Alternatively, alternatively, what is the term for an $n$-dimensional manifold $S$ embedded in Euclidean $n+1$-space such that the geodesic curvature is $0$ at all points in $S$?
i.e. what do you call an $n$-dimensional "flat thing"?
general-topology geometry differential-geometry
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
$endgroup$
What is the term for an $n$-dimensional manifold embedded in Euclidean $n+1$-space for which the function $textbf{p}(t)$ describing the geodesic (parameterized in terms of $t$) between two points has constant partial derivatives ($Vertfrac{d textbf{p}}{dt}Vert=k$ for some real or complex number $k$)? (e.g. a line, plane, etc.)
Alternatively, what is the term for continuous set $S$ of points in Euclidean $n$-space such that there exists a path $P$ between any two points in $S$ described by the funtion $textbf{f}:tto P$, where $Vertfrac{dtextbf{f}}{dt}Vert$ is constant?
Alternatively, alternatively, what is the term for an $n$-dimensional manifold $S$ embedded in Euclidean $n+1$-space such that the geodesic curvature is $0$ at all points in $S$?
i.e. what do you call an $n$-dimensional "flat thing"?
general-topology geometry differential-geometry
general-topology geometry differential-geometry
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
edited Dec 17 '18 at 14:01
R. Burton
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
asked Dec 17 '18 at 4:34
R. BurtonR. Burton
45619
45619
1
$begingroup$
Curves in the plane always have positive curvature; it's the radius of a certain tangent circle. You get negative curvatures in three dimensions.
$endgroup$
– MJD
Dec 17 '18 at 4:39
$begingroup$
A geodesic has constant speed by assumption, so $lVertfrac{mathrm{d}mathbf{p}}{mathrm{d}t}rVert=k$ is trivially true.
$endgroup$
– user10354138
Dec 17 '18 at 14:44
$begingroup$
$textbf{p}(t)$ is the parameterization of the geodesic in Euclidean $n+1$-space, i.e. $textbf{p}(t)=left(x_1(t),x_2(t),ldots,x_n(t)right)$ where $x_1,x_2,ldots,x_n$ are are orthogonal. I'm new to differential-geometry so I might not be saying that right, though.
$endgroup$
– R. Burton
Dec 17 '18 at 14:50
$begingroup$
Are you looking for Hyperplane? If not, how does the thing you describe differ from a hyperplane?
$endgroup$
– MvG
Dec 18 '18 at 0:01
$begingroup$
Yes, that is the thing I was talking about. THANK YOU!
$endgroup$
– R. Burton
Dec 18 '18 at 0:16
add a comment |
1
$begingroup$
Curves in the plane always have positive curvature; it's the radius of a certain tangent circle. You get negative curvatures in three dimensions.
$endgroup$
– MJD
Dec 17 '18 at 4:39
$begingroup$
A geodesic has constant speed by assumption, so $lVertfrac{mathrm{d}mathbf{p}}{mathrm{d}t}rVert=k$ is trivially true.
$endgroup$
– user10354138
Dec 17 '18 at 14:44
$begingroup$
$textbf{p}(t)$ is the parameterization of the geodesic in Euclidean $n+1$-space, i.e. $textbf{p}(t)=left(x_1(t),x_2(t),ldots,x_n(t)right)$ where $x_1,x_2,ldots,x_n$ are are orthogonal. I'm new to differential-geometry so I might not be saying that right, though.
$endgroup$
– R. Burton
Dec 17 '18 at 14:50
$begingroup$
Are you looking for Hyperplane? If not, how does the thing you describe differ from a hyperplane?
$endgroup$
– MvG
Dec 18 '18 at 0:01
$begingroup$
Yes, that is the thing I was talking about. THANK YOU!
$endgroup$
– R. Burton
Dec 18 '18 at 0:16
1
1
$begingroup$
Curves in the plane always have positive curvature; it's the radius of a certain tangent circle. You get negative curvatures in three dimensions.
$endgroup$
– MJD
Dec 17 '18 at 4:39
$begingroup$
Curves in the plane always have positive curvature; it's the radius of a certain tangent circle. You get negative curvatures in three dimensions.
$endgroup$
– MJD
Dec 17 '18 at 4:39
$begingroup$
A geodesic has constant speed by assumption, so $lVertfrac{mathrm{d}mathbf{p}}{mathrm{d}t}rVert=k$ is trivially true.
$endgroup$
– user10354138
Dec 17 '18 at 14:44
$begingroup$
A geodesic has constant speed by assumption, so $lVertfrac{mathrm{d}mathbf{p}}{mathrm{d}t}rVert=k$ is trivially true.
$endgroup$
– user10354138
Dec 17 '18 at 14:44
$begingroup$
$textbf{p}(t)$ is the parameterization of the geodesic in Euclidean $n+1$-space, i.e. $textbf{p}(t)=left(x_1(t),x_2(t),ldots,x_n(t)right)$ where $x_1,x_2,ldots,x_n$ are are orthogonal. I'm new to differential-geometry so I might not be saying that right, though.
$endgroup$
– R. Burton
Dec 17 '18 at 14:50
$begingroup$
$textbf{p}(t)$ is the parameterization of the geodesic in Euclidean $n+1$-space, i.e. $textbf{p}(t)=left(x_1(t),x_2(t),ldots,x_n(t)right)$ where $x_1,x_2,ldots,x_n$ are are orthogonal. I'm new to differential-geometry so I might not be saying that right, though.
$endgroup$
– R. Burton
Dec 17 '18 at 14:50
$begingroup$
Are you looking for Hyperplane? If not, how does the thing you describe differ from a hyperplane?
$endgroup$
– MvG
Dec 18 '18 at 0:01
$begingroup$
Are you looking for Hyperplane? If not, how does the thing you describe differ from a hyperplane?
$endgroup$
– MvG
Dec 18 '18 at 0:01
$begingroup$
Yes, that is the thing I was talking about. THANK YOU!
$endgroup$
– R. Burton
Dec 18 '18 at 0:16
$begingroup$
Yes, that is the thing I was talking about. THANK YOU!
$endgroup$
– R. Burton
Dec 18 '18 at 0:16
add a comment |
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$begingroup$
Curves in the plane always have positive curvature; it's the radius of a certain tangent circle. You get negative curvatures in three dimensions.
$endgroup$
– MJD
Dec 17 '18 at 4:39
$begingroup$
A geodesic has constant speed by assumption, so $lVertfrac{mathrm{d}mathbf{p}}{mathrm{d}t}rVert=k$ is trivially true.
$endgroup$
– user10354138
Dec 17 '18 at 14:44
$begingroup$
$textbf{p}(t)$ is the parameterization of the geodesic in Euclidean $n+1$-space, i.e. $textbf{p}(t)=left(x_1(t),x_2(t),ldots,x_n(t)right)$ where $x_1,x_2,ldots,x_n$ are are orthogonal. I'm new to differential-geometry so I might not be saying that right, though.
$endgroup$
– R. Burton
Dec 17 '18 at 14:50
$begingroup$
Are you looking for Hyperplane? If not, how does the thing you describe differ from a hyperplane?
$endgroup$
– MvG
Dec 18 '18 at 0:01
$begingroup$
Yes, that is the thing I was talking about. THANK YOU!
$endgroup$
– R. Burton
Dec 18 '18 at 0:16