Submanifolds of Kähler and Sasakian Manifolds

In this chapter we study submanifolds in both contact and Kähler geometry. These are extensive subjects in their own right and we give only a few basic results. For a submanifold M of a Riemannian manifold (M̄, g̃) we denote the induced metric by g. Then the Levi-Cività connection ∇ of g and the second fundamental form σ are related to the ambient Levi-Cività connection ∇̃ by % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbae % badaWgaaWcbaGaamiwaaqabaGccaWGzbGaeyypa0Jaey4bIe9aaSba % aSqaaiaadIfaaeqaaOGaamywaiabgUcaRiabeo8aZjaacIcacaWGyb % GaaiilaiaadMfacaGGPaaaaa!4467!$${\bar\nabla_X}Y = {\nabla_X}Y +\sigma(X,Y)$$.