TAILIEUCHUNG - Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold

We give a necessary and sufficient condition for a skew semi-invariant submanifold of order 1 to be a locally warped product. We also establish an inequality between the warping function and the squared norm of the second fundamental form for such submanifolds. The equality case is also discussed. | Turk J Math (2015) 39: 453 – 466 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold Hakan Mete TAS ¸ TAN∗ ˙ ˙ Department of Mathematics, Istanbul University, Istanbul, Turkey Received: • Accepted/Published Online: • Printed: Abstract: We introduce warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold. We give a necessary and sufficient condition for a skew semi-invariant submanifold of order 1 to be a locally warped product. We also establish an inequality between the warping function and the squared norm of the second fundamental form for such submanifolds. The equality case is also discussed. Key words: Locally product manifold, warped product submanifold, skew semi-invariant submanifold, invariant distribution, slant distribution 1. Introduction The theory of submanifolds is a popular research area in differential geometry. In an almost Hermitian manifold, its almost complex structure determines several types of submanifolds. For example, holomorphic (invariant) submanifolds and totally real (anti-invariant) submanifolds are determined by the behavior of the almost complex structure. In the first case, the tangent space of the submanifolds is invariant under the action of the almost complex structure. In the second case, the tangent space of the submanifolds is anti-invariant, that is, it is mapped into the normal space. Bejancu [5] introduced the notion of CR-submanifolds of a K¨ahlerian manifold as a natural generalization of invariant and anti-invariant submanifolds. A CR-submanifold is said to be proper if it is neither invariant nor anti-invariant. The theory of CR-submanifolds has been an interesting topic since then. Slant submanifolds are another generalization of invariant and anti-invariant .

• Toán học
• Locally product manifold
• Warped product submanifold
• Skew semi invariant submanifold
• Invariant distribution
• Slant distribution
• Almost product Riemannian manifold
• Special slant surface
• Pointwise slant submanifolds
• Mean curvature
• Gauss curvature
• Kaehlerian product manifold
• Mixed geodesic submanifold
• Locally decomposable Riemannian manifold
• Constant holomorphic sectional curved manifold
• Riemannian product manifold
• Hemi slant submanifold
• Riemannian manifold
• Parallel canonical structures
• Semi invariant submanifolds
• Riemannian product
• Warped product
• Locally Riemannian product manifold
• Degenerate metric
• Semi Riemannian
• Product manifold
• R lightlike submanifold
• Locally Riemannian product