Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida...
-
Upload
brett-fisher -
Category
Documents
-
view
220 -
download
0
Transcript of Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida...
![Page 1: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/1.jpg)
Bitangencies on Higher Dimensional Immersed Manifolds
Daniel DreibelbisUniversity of North FloridaUSAwww.unf.edu/~ddreibel
![Page 2: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/2.jpg)
Bitengencies on Higher Dimensionel Immersed Menifolds
Deniel DreibelbisUniversity of North FlorideUSEwww.unf.edu/~ddreibel
![Page 3: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/3.jpg)
Outline
Define the problem.Two manifold case.Transitions in bitangencies.One manifold case.Second and third order geometry.Tangent translations.Non-tangent translations.Putting it all together.
![Page 4: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/4.jpg)
Line Bitangencies
![Page 5: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/5.jpg)
Curve and Surface
Fabricius-Bjerre and Halpern
O - S + D + ½ I = 0
![Page 6: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/6.jpg)
General Case: Two manifolds
![Page 7: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/7.jpg)
Changes in Bitangencies
![Page 8: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/8.jpg)
Annoying Transition
![Page 9: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/9.jpg)
One Manifold Case
![Page 10: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/10.jpg)
Asymptotic Vectors
![Page 11: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/11.jpg)
Asymptotic Vectors
2-manifold
Hyperbolic point Elliptic point
![Page 12: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/12.jpg)
Asymptotic Vectors
3-manifold
![Page 13: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/13.jpg)
Asymptotic Vectors
4-manifold
![Page 14: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/14.jpg)
Parabolic Set
2-D 3-D
![Page 15: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/15.jpg)
Calculating I
![Page 16: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/16.jpg)
Singularities at a Parabolic Point
![Page 17: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/17.jpg)
Singularities at a Parabolic Point
![Page 18: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/18.jpg)
The Manifold and its Translate
![Page 19: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/19.jpg)
Generic Condition on the Translation
![Page 20: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/20.jpg)
Tangent Translations
![Page 21: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/21.jpg)
Conjugate Planes
![Page 22: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/22.jpg)
Signs of the Bitangencies
![Page 23: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/23.jpg)
Analyzing the Conjugate Plane
2-D hyperbolic 3-D manifold
![Page 24: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/24.jpg)
Nontangent Translations
![Page 25: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/25.jpg)
Flecnodal Normal
![Page 26: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/26.jpg)
Final Formula
![Page 27: Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA ddreibel.](https://reader036.fdocuments.us/reader036/viewer/2022062305/56649f505503460f94c72a64/html5/thumbnails/27.jpg)
Thenk you!!
www.unf.edu/~ddreibel