Differential geometry of warped product manifolds and submanifolds v 0. Geometry of warped product and crwarped product submanifolds in kaehler manifolds. Geometry of warped product and cr warped product submanifolds in kaehler manifolds. In 26, it was proved that there is no warped product semislant submanifold of the form m t. Let b,g b and f,g f be two riemannian manifolds with riemannian metric g b and g f, respectively, and fa positive di. Apr 10, 2017 in this paper, we study semislant submanifolds and their warped products in kenmotsu manifolds.
But we show by means of examples the existence of warped product semislant submanifolds such that the totally geodesic submanifold of a warped product is a proper slant submanifold in locally riemannian product manifolds. Some characterizing results for hemislant warped product. The existence of warped product hemislant submanifolds of an lpsasakian manifold is also ensured by an interesting example. Jul 17, 2017 a warped product manifold is a riemannian or pseudoriemannian manifold whose metric tensor can be decomposed into a cartesian product of the y geometry and the x geometry except that the xpart is warped, that is, it is rescaled by a scalar function of the other coordinates y. Thus, the geometry of warped product slant submanifolds with indenite metric became a topic of investigation. This site is like a library, use search box in the widget to get ebook that you want. Geometry of warped product pointwise semislant submanifolds. Warped products in sasakian manifolds differential.
Second, we study statistical warped products as submanifolds of statistical manifolds. Contact crsubmanifolds of sasakian manifolds were introduced by yano and. Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. During this period, ecommerce and registration of new users may not be available for up to 12 hours. Buy differential geometry of warped product manifolds and submanifolds on free shipping on qualified orders. You can consider warped products of manifolds of arbitrary dimensions from a purely geometrical viewpoint, and apply a more general theory in all of these particular cases. Later on, many geometers studied such type of warped product submanifolds in almost hermitian as well as almost contact manifolds. Hasegawa and mihai 19and munteanu 26 studied the warped product contact crsubmanifolds in sasakian manifolds. Another characterization of warped product submanifolds of. Warped product manifolds have been studied for a long period of time. Geometry of warped product and crwarped product submanifolds. The geometry of warped products has a crucial role in differential geometry, as well as physical. Geometric inequality of warped product semislant submanifolds of. Dec 30, 2008 we provide a general study of submanifolds in r m k, f.
He has established a sharp relationship between the squared norm of the second fundamental form and warping function. The existence of such warped products in kenmotsu manifolds is shown by an example and a characterization. We obtain the results on the nonexistance or existence of warped product hemislant submanifolds and give some examples of lpsasakian manifolds. Differential geometry of submanifolds of warped product.
Geometry of warped product semislant submanifolds of. Although there are many papers concerning the geometry of semiinvariant submanifolds of almost paracontact riemannian manifolds see 11, there is no paper concerning the geometry of warped product semiinvariant submanifolds of almost paracontact riemannian manifolds in literature so far. Pdf differential geometry of warped product manifolds and. Pdf warped product pointwise bislant submanifolds of. As a generalization of warped product cr submanifolds warped product semislant submanifolds are very important in differential geometry. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle.
Fundamental properties of submanifolds in r m k, f are obtained. We provide a general study of submanifolds in r m k, f. We study of warped product submanifolds, especially warped product hemislant submanifolds of lpsasakian manifolds. The notion of slant submanifolds of almost hermitian manifolds was. Recently, chenmunteanu, brought our attention to the geometry of pr warped products in parak ahler. Geometry of warped products as riemannian submanifolds and related problems by bangyen chen abstract.
Similarly, a symmetric bilinear form b is called negative definite. Warped products in sasakian manifolds differential geometry. Pdf we provide a general study of submanifolds in rmk, f. The flrw model, hyperbolic space, and surfaces of revolution can also be represented as warped products.
First, we suppose that n1 is an invariant and n2 is a semislant of m with slant angle. Differential geometry held at tokyo metropolitan university, december 1719, 2001. Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research, and many articles are available. Thanks for contributing an answer to mathematics stack exchange.
Easiest examples of warped product manifolds are surfaces of revolution. The geometry of submanifolds download ebook pdf, epub. Another inequality for contact cr warped products in sasakian manifolds. Ozgur studied einstein statistical warped product manifolds. Some basic inequalities for submanifolds of nearly quasiconstant curvature manifolds, pp. Differential geometry of submanifolds of warped product manifolds i. Sahin 10 studied and obtained the wintgenlike inequality for statistical submanifolds of statistical warped product manifolds. Yun kyong kim and dong ho lim characterizations of real hypersurfaces with structure lie operator in a nonflat complex space form.
Semiinvariant warped product submanifolds of almost contact. Geometry of warped product manifolds10419 wsbook9x6 page xxviii xxviii di. Differential geometry of warped product manifolds and submanifolds, pp. Warped product semiinvariant submanifolds in almost. Hemislant warped product submanifolds of nearly kaehler manifolds fallehr. In this paper, we study semislant submanifolds and their warped products in kenmotsu manifolds. A warped product manifold is a riemannian or pseudoriemannian manifold whose metric tensor can be decomposed into a cartesian product of the y geometry and the x geometry except that the xpart is warped, that is, it is rescaled by a scalar function of the other coordinates y. Pr pseudoslant warped product submanifold of a nearly. Such notion plays very important roles in differential geometry as well as in physics, especially in general relativity. As a generalization of warped product crsubmanifolds warped product semislant submanifolds are very important in differential geometry. Hasegawa and mihai 19 and munteanu 26 studied the warped product contact cr submanifolds in sasakian manifolds. Warped product submanifolds in metallic riemannian manifolds. With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on a warped product manifolds for instance and warped product manifolds are widely used in differential geometry, physics and as well as in different branches of engineering. But avoid asking for help, clarification, or responding to other answers.
Several classification theorems on parallel, curvatureinvariant and totally umbilical submanifolds in r m k, f are proved. Click download or read online button to get the geometry of submanifolds book now. Geometric inequality of warped product semislant submanifolds of locally product riemannian manifolds. Viqar azam khan and kamran khan semislant warped product submanifolds of a nearly kaehler manifold, pp. In the available literature,many geometers have studied warped products in the setting of almost contact met ric manifolds c.
Applications of the symmetric criticality principal in mathematical physics and differential geometry, proc. Research article hemislant warped product submanifolds of. Noncommutative geometry edit for a c k manifold m, the set of realvalued c k functions on the manifold forms an algebra under pointwise addition and multiplication, called the algebra of scalar fields or simply. The geometry almost decomposes into a cartesian product of the y geometry and the x geometry except that the x part is warped, i. Pdf geometry of biwarped product submanifolds in sasakian. Pdf differential geometry of submanifolds of warped product. Warped product einstein manifolds and hessian pde, with a. Pdf differential geometry of warped product manifolds. Differential geometry of submanifolds of warped product manifolds. In this paper, we study the existence of proper warped product submanifolds in metallic or golden riemannian manifolds and we discuss about semiinvariant, semislant and, respectively, hemislant warped product submanifolds in metallic and golden riemannian manifolds. Later, n1 will be an antiinvariant submanifold and n2 will be a semi. A survey on geometry of warped product submanifolds.
This generalized product metric appears in differential geometric studies in a. Alsolamy 1 andmerajalikhan 2 department of mathematics, king abdulaziz university, p. For this reason, the metric of a warped geometry is often called a warped product metric. Geometry of crsubmanifolds connecting repositories. Differential geometry abstract in this report, we consider doubly warped product manifolds and we get fundamental properties of this manifold and consider these submanifolds. In this section, we study warped product semislant submanifolds, with warped product in the form n n1. Box, jeddah, saudi arabia department of mathematics, university of tabuk, tabuk, saudi arabia correspondence should be addressed to meraj ali khan. A warped product manifold is a riemannian or pseudoriemannian manifold whose metric tensor can be decomposes into a cartesian product of the y geometry and the x geometry except that the x. Characterization of gcrlightlike warped product of indefinite. During this period, e commerce and registration of new users may not be available for up to 12 hours. Oct 29, 2004 differential geometry abstract in this report, we consider doubly warped product manifolds and we get fundamental properties of this manifold and consider these submanifolds.
In contrast, the study of warped product submanifolds from extrinsic point of view was initiated by the first author around the beginning of this century in 7, 8. Differential geometry of warped product manifolds and submanifolds bangyen chen a warped product manifold is a riemannian or pseudoriemannian manifold whose metric tensor can be decomposed into a cartesian product of the y geometry and the x geometry except that the xpart is warped, that is, it is rescaled by a scalar function of the. Since then the study of warped product submanifolds has been investigated by many geometers. Differential geometry of warped product manifolds and submanifolds. Differential geometry of warped product manifolds and. Statistical solitons and inequalities for statistical warped. Motivated by the work of 11, the authors have studied pr. Notes on submanifolds in warped products sciencedirect. Many differential geometric properties of submanifolds of a kaehler manifold are looked into via canonical structure tensors p and f on the submanifold. The book also contains material on the general theory of connections on vector bundles and an indepth chapter on semiriemannian geometry that covers basic material about riemannian manifolds and lorentz manifolds. Pointwise bislant and hemislant warped products in sasakian manifolds. The geometry of warped product submanifolds of a locally product riemannian manifold is quite different from the geometry of warped products in a kaehler manifold.
Warped product submanifolds of lpsasakian manifolds. Warped product submanifolds of riemannian product manifolds. Warped products play very important roles in differential geometry as well as in physics. Also, we provide some examples of warped product submanifolds in euclidean spaces. In this paper, we study submanifolds in the warped product m. Warped product submanifolds of lorentzian paracosymplectic. The notion of warped product manifolds plays very important. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in euclidean space.
128 1495 288 372 1501 847 272 547 1459 155 34 1514 1369 1302 678 1501 641 957 539 600 980 1106 1269 138 1072 619 805 1258 1399 1133 559 1262 557 632 362 831