On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold
1985 ◽
Vol 28
(3)
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pp. 305-311
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Keyword(s):
Let Mn be an n-dimensional smooth compact Riemannian manifold. By a theorem of Nash, we can think of it as an isometrically immersed submanifold in some higher dimensional Euclidean space ℝn+m. Viewing in this way we can compare the intrinsic geometry of M to its extrinsic geometry. Classically, the Gauss equationwhere K(X,Y) denotes the sectional curvature in M corresponding to the plane spanned by the two orthonormal vectors X, Y and B denotes the second fundamental form gives one of the most important relations between the intrinsic and extrinsic geometries of M. In this note we shall prove the following.
2000 ◽
Vol 24
(1)
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pp. 43-48
1974 ◽
Vol 29
(11)
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pp. 1527-1530
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2002 ◽
Vol 132
(5)
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pp. 1163-1183
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2010 ◽
Vol 21
(11)
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pp. 1421-1428
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2002 ◽
Vol 132
(5)
◽
pp. 1163-1183
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1975 ◽
Vol 27
(3)
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pp. 610-617
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1996 ◽
Vol 126
(6)
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pp. 1217-1234
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1981 ◽
Vol 23
(2)
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pp. 249-253
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