Differential Geometry (Research Notes In mathematics Series) by Luis A. Cordero PDF

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For If Ø:M + is a map between two Riemannian manifolds (I't,g) and (N,h), we N define: (I) its first fundamental (ii) its second fundamental form 4*h, a syninetric 2-covariant tensor field on a syninetric 2-covariant tensor field on II with values in the induced tangent bundle p1TN. form = In local charts, * = ij "ij (B(A))Y + c*B' subscripts on q denoting covariant derivatives. 46 We are interested in various variational principles associated with inteintegration with respect to the grals of the form = J volume element Vg over compact domains of N.

N.. ,n. 0 for any normal to N. 3), N is totally geodesic. 18) we have 0 = for any 'v. Thus, B is normal to 'v. ,n. ,n, so that B is normal to J(TN), and therefore it must be tangent to N. Since n 5. 8). Put vu(PV) - for all U,V tangent to N. 1) = vu(PV) - + — + = 0. K. manifold V is called a CRa holomorphic submanifold product T N and a totally real submanifold m of V. K. manifold V such that the Lee field B is normal to N. 2 parallel. 2) we have = V'P. 4) (FV)U. Fnr any X 0, Fr. 4) gives + Q or, equivalently, g(c(U,X),JV) + = 0, for any V tangent to N.

Manifold V such that for any x N. Then N is locally the Riemannian product of a submanifold and a totally reel submanifold of V. 3 N is locally the product of a. holomorphic submanifold and a totally real submanifold N1 of V. Since w 0 on N, we have induced structure on NT. Moreover, it can easily be seen that the pro- Proof a jection map p (or q) onto D CD1) is parallel with respect to local product is actually a local Rieniannian product. Next, we consider the particular case when N is that this holomorphic or 'totally real.

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