2007 Fiscal Year Final Research Report Summary
Geometric structures and submanifolds
| Project/Area Number |
17540078
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Shimane University |
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Principal Investigator |
KIMURA Makoto Shimane University, Fac. of Sci. and Tech, Professor (30186332)
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| Co-Investigator(Kenkyū-buntansha) |
FURUMOCHI Tetsuo Shimane Univ., Fac. of Sci. and Tech, Professor (40039128)
HATTORI Yasunao Shimane Univ., Fac. of Sci. and Tech, Professor (20144553)
YOKOI Katsuya Shimane Univ., Fac. of Sci. and Tech, Professor (90240184)
MAEDA Sadahiro Saga Uni., Fac. of Sci. and Tech, Professor (40181581)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Differential Geometry / Lagrange submanifolds / Minimal submanifolds |
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| Research Abstract |
As a joint research with Kaoru Suizu, we obtained fundamental theorem for Lagrange surfaces in the Riemann product of round 2-spheres. Namely, we showed that for minimal Lagrange surfaces in $S^2\times S^2$, Gauss and Codazzi equations are sufficient and necessary condition far the existence of such minimal Lagrange immersions. Also for Lagrange surfaces in $S^2\times S^2$, second fundamental tensor and the angle function, which we introcuced are invariant for congruent.
Next we investigated Lagrange submanifolds in complex projective spaces , which is obtained as a 1-parameter family of totally geodesic, totally real (n-1)-dimensional submanifolds $RP^<{n-1}>$. Such submanifolds axe obtained from curves in the moduli space op $RP^<{n-1}>$ in $CP^n$. For a curve in the moduli space, we showed that the corresponding $n$-dimensional submanifold is Lagrangian if only if the curve is horizon with respect to the natural fibration from the moduli space to $CP^n$. Then we showed that if such a Lagrange submanifold is minimal, then is total geode-sic. More we investigated the condition for which such Lagrange submanifolds to be Hamiltonian minimal.
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