Parallel submanifolds of complex projective space and their normal holonomy sergio console and antonio j. Discrete groups, symmetric spaces, and global holonomy authors. In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Homogeneity and normal holonomy article pdf available in bulletin of the london mathematical society 416. For a map of a closed surface f g, the curvature is zero on the 2manifold in this case the holonomy evaluated on the fundamental cycle of. Submanifolds and holonomy 2nd edition 9781482245158. Manifolds with special holonomy and their calibrated. Title the normal holonomy of crsubmanifolds authors di. The proof uses euclidean submanifold geometry of orbits and gives a link between.
Submanifolds, holonomy, and homogeneous geometry request. Calibrated submanifolds naturally arise when the ambient manifold has special holonomy, including holonomy g2. The exceptional holonomy groups and calibrated geometry. Parallel submanifolds of complex projective space and their.
It concerns calibrated submanifolds, a special kind of minimal submanifold of a. It also adds more than 100 references to the bibliography and substantially improves the index. Gbecause tis abelian, so the gerbe is flat there, but the holonomy is nonzeroit is a rather subtle mod 2 invariant of the group. The high dimensional holonomy map for ruled submanifolds 6. A submanifold of a riemannian manifold is called an extrinsic sphere if it is totally. Indeed, we explain how these submanifolds can be regarded as the unique complex.
We will also survey on recent results obtained in cooperation with j. Olmos holonomy groups and applications in string theory universitat hamburg july 14 18, 2008. This second edition includes five new chapters on the normal holonomy of complex submanifolds, the bergersimons holonomy theorem, the skewtorsion holonomy theorem, and polar actions on symmetric spaces of compact type and noncompact type. Calibrated submanifolds clay mathematics institute. Parallel submanifolds of complex projective space and. Proceedings of the seventh international workshop on differential geometry, daegu, november 1516, 2002 real and complex submanifolds lab. Its calibrated submanifolds are called special lagrangian mfolds, or sl mfolds for short.
The proof uses euclidean submanifold geometry of orbits and gives a link between riemannian holonomy groups and normal holonomy groups. This gives a characterization of veronese submanifolds in terms of normal holonomy. Submanifolds and holonomy 2nd edition by jurgen berndt and publisher chapman and hallcrc. We use cookies to make interactions with our website easy and meaningful, to better understand the use of our services, and to tailor advertising.
Totally geodesic submanifolds of regular sasakian manifolds murphy, thomas, osaka journal of. A berger type normal holonomy theorem for complex submanifolds. Normal holonomy theorem is a very important tool for the study of submanifold geometry, especially in the context of submanifolds with simple extrinsic geometric invariants, like isoparametric and homogeneous submanifolds see 6 for an introduction to this subject. A geometric proof of the berger holonomy theorem by carlos olmos dedicated to ernst heintze on the occasion of his sixtieth birthday abstract we give a geometric proof of the berger holonomy theorem. Normal holonomy groups, for riemannian submanifolds of euclidean. The deformation problem is elliptic but in general obstructed. It turns out that the normal holonomy group is compact but it does not act, in general. Olmos sergio console july 14 18, 2008 contents 1 main results 2 2 submanifolds and holonomy 2. The exceptional holonomy groups and calibrated geometry a g2structure on a 7manifold mis a principal subbundle of the frame bundle of m, with structure group g2. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the projectivized isotropy.
Mclean studied the deformations of closed cayley submanifolds. Discrete groups, symmetric spaces, and global holonomy. Pdf homogeneity of infinite dimensional antikaehler. Jan 22, 2008 the object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We study the uniqueness of minimal submanifolds and the stability of the mean curvature ow in several wellknown model spaces. Deformations of calibrated submanifolds 709 deformation of submanifolds, ft. This second edition includes five new chapters on the normal holonomy of complex. Submanifolds and holonomy, second edition explores recent progress in the submanifold geometry of space forms, including new methods based on the holonomy of the normal connection.
For flat connections, the associated holonomy is a type of monodromy and is an inherently. Special lagrangian mfolds in calabiyau mfolds with holonomy sum, and realdimension 2m. Normal holonomy and rational properties of the shape operator. Complex submanifolds and holonomy sergio console main results.
We prove a berger type theorem for the normal holonomy i. Submanifolds, holonomy, and homogeneous geometry springerlink. There are different types of submanifolds depending on exactly which properties are required. Submanifolds and holonomy, second edition by jurgen berndt 2016 english pdf. Pdf a berger type normal holonomy theorem for complex. Apr 28, 2003 with special emphasis on new techniques based on the holonomy of the normal connection, this book provides a modern, selfcontained introduction to submanifold geometry. Let g be a lie group and p a principal g bundle over a smooth manifold m which is paracompact. Submanifolds and subbundles we consider a submanifold m. This second edition reflects many developments that have occurred since the publication of its popular predecessor. Submanifolds and holonomy jurgen berndt, sergio console. For totally real submanifolds of totally geodesic totally real submanifolds of a complex space form we give an explicit description of the action of its normal holonomy group. It is shown that a submanifold of euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifold.
In this situation, we would hope that the calibrated submanifolds encode even more. Use features like bookmarks, note taking and highlighting while reading submanifolds and holonomy chapman. This book provides an introduction to submanifold geometry with a special emphasis on new techniques based on the holonomy of the normal connection. Thirdly, there is a tubular neighborhood of x in m that is identified via the normal exponential. The normal holonomy of cr submanifolds 3 results about reduction of codimension. Homogeneity of infinite dimensional antikaehler isoparametric submanifolds ii. Download it once and read it on your kindle device, pc, phones or tablets. Holonomy holonomy and curvature holonomy and tensor.
Submanifolds, holonomy, and homogeneous geometry request pdf. Complex submanifolds and holonomy joint work with a. Pdf we give a geometric proof of the berger holonomy theorem. Introduction the goal of this work is to study the deformability of a some particular kind of submanifolds immersed in an equiregular graded manifold n,h1. Submanifolds and holonomy 2nd edition jurgen berndt. Geometry of g2 orbits and isoparametric hypersurfaces miyaoka, reiko, nagoya mathematical journal, 2011.
Different authors often have different definitions. The normal holonomy of crsubmanifolds 5 we will now introduce some preliminaries on the general theory of sub manifolds of a complex space form and state how the geometry of a submanifold mof the complex projective or hyperbolic space relates with that of if pullback via the hopf bration. Stability of certain reflective submanifolds in compact symmetric spaces kimura, taro, tsukuba journal of mathematics, 2008. X xt c m may be assumed to be a normal deformation, i. This is an expository article about our joint published research with sergio. Riemannian holonomy groups and calibrated geometry pdf riemannian holonomy groups and calibrated geometry. Normal holonomy of orbits and veronese submanifolds olmos, carlos and rianoriano, richar, journal of the mathematical society of japan, 2015. Manifolds with special holonomy and their calibrated submanifolds and connections bobby acharya kings college london ictp robert bryant uc berkeley msri spiro karigiannis university of waterloo naichung conan leung chinese university of hong kong ims sunday, 29042012 to friday, 04052012 1 overview of the. We study the uniqueness of minimal submanifolds and the stability of the mean curvature ow in several wellknown model spaces of manifolds of special holonomy. Riemannian holonomy groups and calibrated geometry pdf. Publications of jurgen berndt kings college london.
Discrete groups, symmetric spaces, and gl,obal holonomy. Pdf submanifolds with constant principal curvatures and normal. Let m be a connected simply connected riemannian manifold and let r be a properly discontinuous group of isometries such that. Mean curvature flows in manifolds of special holonomy chungjun tsai and mutao wang abstract. Pdf a geometric proof of the berger holonomy theorem.
With special emphasis on new techniques based on the holonomy of the normal connection, this book provides a modern, selfcontained introduction to submanifold geometry. On counting associative submanifolds and seibergwitten monopoles. Simons collaboration on special holonomy in geometry, analysis and physics home page spaces with special holonomy are of intrinsic interest in both mathematics and mathematical physics. For riemannian manifolds there are four kinds of holonomy groups. Riemannian manifold m, which are defined using a closed form on m called a.
Submanifolds and holonomy, second edition explores recent progress in the submanifold geometry of space forms, including new methods based on the. We prove that a maximal totally complex submanifold n2n of the quaternionic projective space n n. It offers a thorough survey of these techniques and their applications and presents a framework for various recent results to date found only in scattered research papers. Request pdf submanifolds, holonomy, and homogeneous geometry this is an expository article.
On counting associative submanifolds and seibergwitten. The definition for holonomy of connections on principal bundles proceeds in parallel fashion. This is because, if x is compact, then one can reparametrize using a time depen dent diffeomorphism of x. Introduction to riemannian holonomy groups and calibrated. A submanifold has by definition constant principal curvatures if the eigenvalues of the shape operators a.
Simons collaboration on special holonomy in geometry. These are constructed and studied using complex algebraic. In mathematics, a submanifold of a manifold m is a subset s which itself has the structure of a manifold, and for which the inclusion map s m satisfies certain properties. We would like to draw the attention to some problems in submanifold and homogeneous geometry related to the socalled normal holonomy. The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Save up to 80% by choosing the etextbook option for isbn. We would like to draw the attention to some problems in submanifold and homogeneous geometry related. If this 4form phi is closed, then the holonomy of m is contained in spin7 and cayley submanifolds are calibrated minimal submanifolds. Pdf a submanifold has by definition constant principal curvatures if the eigenvalues of. Submanifolds and holonomy 2nd edition jurgen berndt sergio. It is shown that a submanifold of euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifo. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality in the sphere.
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