Parallel hypersurfaces in a riemannian manifold and focal points
$begingroup$
For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $$S_t={exp^perp(v):vin T(S)^perp,;|v|=t}$$ and $$f_t:Srightarrow S_t:pmapsto exp^perp(teta)$$ with $eta$ the unit normal vector, then what is the relation of $f_t$ and the focal points of a geodesic $gammaperp S$?
We say that $f_t$ is a diffeomorphism for small $t$. How can we determine the value of $t$ where it stops being a diffeomorphism?
Any references on this material?
differential-geometry riemannian-geometry
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
$endgroup$
add a comment |
$begingroup$
For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $$S_t={exp^perp(v):vin T(S)^perp,;|v|=t}$$ and $$f_t:Srightarrow S_t:pmapsto exp^perp(teta)$$ with $eta$ the unit normal vector, then what is the relation of $f_t$ and the focal points of a geodesic $gammaperp S$?
We say that $f_t$ is a diffeomorphism for small $t$. How can we determine the value of $t$ where it stops being a diffeomorphism?
Any references on this material?
differential-geometry riemannian-geometry
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
$endgroup$
add a comment |
$begingroup$
For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $$S_t={exp^perp(v):vin T(S)^perp,;|v|=t}$$ and $$f_t:Srightarrow S_t:pmapsto exp^perp(teta)$$ with $eta$ the unit normal vector, then what is the relation of $f_t$ and the focal points of a geodesic $gammaperp S$?
We say that $f_t$ is a diffeomorphism for small $t$. How can we determine the value of $t$ where it stops being a diffeomorphism?
Any references on this material?
differential-geometry riemannian-geometry
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
$endgroup$
For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $$S_t={exp^perp(v):vin T(S)^perp,;|v|=t}$$ and $$f_t:Srightarrow S_t:pmapsto exp^perp(teta)$$ with $eta$ the unit normal vector, then what is the relation of $f_t$ and the focal points of a geodesic $gammaperp S$?
We say that $f_t$ is a diffeomorphism for small $t$. How can we determine the value of $t$ where it stops being a diffeomorphism?
Any references on this material?
differential-geometry riemannian-geometry
differential-geometry riemannian-geometry
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
edited Jan 10 '15 at 5:05
Mark Fantini
4,88041936
4,88041936
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
asked Nov 17 '14 at 22:00
Test123Test123
2,792828
2,792828
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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My uneducated view (disposable if a real answer is posted) is that I think we can continue this process right until a focal point is reached, of the apparently discrete set of such points (references therein might be helpful too). Consider the unit sphere, we start from the equator to parallel disks at each latitude up until the north pole when $ t = 1 $.
edited Jan 9 at 7:03
El borito
664216
answered Jan 10 '15 at 4:48
rychrych
2,5161718
$endgroup$
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
My uneducated view (disposable if a real answer is posted) is that I think we can continue this process right until a focal point is reached, of the apparently discrete set of such points (references therein might be helpful too). Consider the unit sphere, we start from the equator to parallel disks at each latitude up until the north pole when $ t = 1 $.
edited Jan 9 at 7:03
El borito
664216
answered Jan 10 '15 at 4:48
rychrych
2,5161718
$endgroup$
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
add a comment |
$begingroup$
My uneducated view (disposable if a real answer is posted) is that I think we can continue this process right until a focal point is reached, of the apparently discrete set of such points (references therein might be helpful too). Consider the unit sphere, we start from the equator to parallel disks at each latitude up until the north pole when $ t = 1 $.
edited Jan 9 at 7:03
El borito
664216
answered Jan 10 '15 at 4:48
rychrych
2,5161718
$endgroup$
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
add a comment |
$begingroup$
My uneducated view (disposable if a real answer is posted) is that I think we can continue this process right until a focal point is reached, of the apparently discrete set of such points (references therein might be helpful too). Consider the unit sphere, we start from the equator to parallel disks at each latitude up until the north pole when $ t = 1 $.
edited Jan 9 at 7:03
El borito
664216
answered Jan 10 '15 at 4:48
rychrych
2,5161718
$endgroup$
My uneducated view (disposable if a real answer is posted) is that I think we can continue this process right until a focal point is reached, of the apparently discrete set of such points (references therein might be helpful too). Consider the unit sphere, we start from the equator to parallel disks at each latitude up until the north pole when $ t = 1 $.
edited Jan 9 at 7:03
El borito
664216
answered Jan 10 '15 at 4:48
rychrych
2,5161718
edited Jan 9 at 7:03
El borito
664216
edited Jan 9 at 7:03
El borito
664216
edited Jan 9 at 7:03
El borito
664216
664216
answered Jan 10 '15 at 4:48
rychrych
2,5161718
answered Jan 10 '15 at 4:48
rychrych
2,5161718
answered Jan 10 '15 at 4:48
rychrych
2,5161718
2,5161718
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
add a comment |
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
$begingroup$
Focal points are only one part of the story, they detect failure of the normal exponential map to be an immersion. However, an immersion need not be a diffeomorphism to its image because it fails to be injective. As an example, consider $n=2$, $S$ a simple closed geodesic and $M$ compact of negative curvature. There are no focal points in this situation, but, $f_t$ will fail to be injective for large $t$.
$endgroup$
– Moishe Kohan
Jan 7 at 18:08
add a comment |
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