• East Asian mathematical journal
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• v.14 no.2
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• pp.269-279
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• 1998

For a codimension 1 submanifold in a Euclidean 2n-space, the degree of the gauss map mod 2 is the semi-characteristic of the manifold in $Z_2$ coefficient.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.289-297
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• 2002

It was shown by L. Polterovich ([3]) that if L is a totally real submanifold of a symplectic manifold $(M,\omega)$ and L is parallelizable then L is normal. So we try to find an answer to the question of whether there is a compatible almost complex structure J on the symplectic vector bundle $TM$\mid$_{L}$ such that $TL{\cap}JTL=0$ assuming L is normal and parallelizable. Although we could not reach an answer, we observed that the claim holds at the vector space level. And related to the question, we showed that for a symplectic vector bundle $(M,\omega)$ of rank 2n and $E=E_1{\bigoplus}E_2$, where $E=E_1,E_2$are Lagrangian subbundles of E, there is an almost complex structure J on E compatible with ${\omega}$ and $JE_1=E_2$. And finally we provide a necessary and sufficient condition for a given embedding into a symplectic manifold to be normal.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.183-189
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• 2013

Let W be a parallelizable compact oriented manifold of dimension $n$ with boundary ${\partial}W=M$. We define the so-called Gauss map $f:M{\rightarrow}S^{n-1}$ using a framing of TW and show that the degree of $f$ is equal to Euler-Poincar$\acute{e}$ number ${\chi}(W)$, regardless of the specific framing. As a special case, we get a Hopf theorem.