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Umehara Lab
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A differential geometry lab that studies the shapes of curves and surfaces.
AboutAbout the lab
Studying visual objects such as curves and surfaces.
The Umehara Lab studies differential geometry.
Its research subjects are visual objects such as curves and surfaces.
Current work includes maximal surfaces in spacetime, surfaces of constant mean curvature 1 in 3-dimensional de Sitter space, and isometric deformations of singularities.
The lab is also interested in the index of umbilic points of surfaces, and continues to study it as a life's work.
Key terms
- Differential geometry
- A field of mathematics that investigates how curves and surfaces bend, using the ideas of calculus.
- Singularity
- A point where a surface or similar object ceases to be smooth; a place where it is pointed or folded over.
- Umbilic point
- A point on a surface that curves by the same amount in every direction. The Japanese name (saiten) means "navel."
ResearchResearch themes
From the standpoint of differential geometry, the lab studies the shapes of curves and surfaces. Its subjects include singularities of surfaces and hypersurfaces, flat tori in the 3-dimensional sphere, and maximal surfaces in spacetime.
*Illustrative imageInvestigating the shapes of pointed surfaces
Like the wavefronts of light, surfaces can develop sharp edges and singularities called "swallowtails." The lab studies formulas describing the shape around such singularities, and how the singularities change when a surface is gradually deformed.
*Illustrative imageSurfaces in spacetime
In Lorentz-Minkowski space, which represents the spacetime of relativity, the properties of surface shapes differ from those in ordinary space. The lab studies umbilic points of surfaces in this space, surfaces with zero mean curvature, and timelike surfaces.
MembersFaculty
- ProfessorMasaaki UmeharaPublications ↗
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Catalan minimal surface 03” Rectas / CC0 (cropped)
- Investigating the shapes of pointed surfaces: “Quartic Discriminant” Salix alba / CC BY-SA 4.0 (cropped)
- Surfaces in spacetime: “Light cone cs” Timichal / Public domain (cropped)
- Related lab (土岡俊介研究室): “Gosset 1 22 polytope” Tom Ruen / Public domain (cropped)
- Related lab (鹿島亮研究室): “Formulas on an old blackboard (Unsplash)” Roman Mager roman_lazygeek / CC0 (cropped)
- Related lab (西畑伸也研究室): “Collision de solitons hydrodynamiques” Christophe.Finot, Julien Fatome, Stéphane Pitois / CC BY-SA 4.0 (cropped)
- Related lab (高橋仁研究室): “Heat” Nicoguaro. Based on File:Heat eqn.gif by en:User:Oleg Alexandrov / CC BY 4.0 (cropped)
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