By Alfred Barnard Basset

Initially released in 1910. This quantity from the Cornell collage Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 layout by way of Kirtas applied sciences. All titles scanned hide to hide and pages could contain marks notations and different marginalia found in the unique quantity.

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22) and Codazzi equation g(R(ıX, ıY )ıZ, ξa ) = g (∇X Aa )Y − (∇Y Aa )X, Z p sba (X)g(Ab Y, Z) − sba (Y )g(Ab X, Z) . 23) b=1 After computing R(ıX, ıY )ξa in the same way, we obtain R(ıX, ıY )ξa = ∇X ∇Y ξa − ∇Y ∇X ξa − ∇[X,Y ] ξa = ı −(∇X Aa )Y + (∇Y Aa )X − (sab (Y )Ab X − sab (X)Ab Y ) b (∇X sab )(Y ) − (∇Y sab )(X) − g((Aa Ab − Ab Aa )X, Y ) + b [sac (Y )scb (X) − sac (X)scb (Y )] ξb , + c and we deduce the following Ricci-K¨ uhne equation: g(R(ıX, ıY )ξa , ξb ) = g (Ab Aa − Aa Ab )X, Y + (∇X sab )(Y ) − (∇Y sab )(X) sac (Y )scb (X) − sac (X)scb (Y ) .

An n-dimensional totally geodesic submanifold M of (n + p)dimensional Euclidean space En+p is an open submanifold of n-dimensional Euclidean space. If M is complete, then M is an n-dimensional Euclidean space. Proof. 27) implies R⊥ (X, Y )ξa = 0, that is, the normal curvature vanishes identically. 5, we can choose orthonormal normal vector ﬁelds ξ1 , . . , ξp to M in such a way that the corresponding third fundamental form will vanish. Therefore, ∇X ξa = 0, since M is a totally geodesic submanifold.

En be an orthonormal basis of Tx (M ) such that e1 , . . , en−1 ∈ Hx (M ). Then en ∈ Tx (M ) \ Hx (M ) and g(Jıen , ıej ) = −g(ıen , Jıej ) = 0, j = 1, . . , n − 1, since Jıej ∈ Hx (M ). 8. 3 are equivalent. 7 is false. Let M be a CR submanifold of CR dimension n−2 2 . Choosing an orthonormal basis e1 , e2 , . . , en−2 , en−1 , en of Tx (M ) in such a way that e1 , e2 , . . , en−2 ∈ JTx (M ) ∩ Tx (M ), 7 Submanifolds of a complex manifold 47 we can write Jei ∈ JTx (M ) ∩ Tx (M ), i = 1, .