orbits of rank one and parallel mean curvature let mn n 2 be a extrinsic homogeneous irreducible full submanifold of euclidean space with rank m k 1 i e it admits k 1 locally defined linearly independent parallel normal vector fields we prove that m must be contained in a sphere this result toghether with previous work of the author about homogeneous submanifolds of higher rank imply in particular the following theorem a homogeneous irreducible submanifold of euclidean space with parallel mean curvature vector is either minimal or minimal in a sphere or an orbit of the isotropy representation of a simple symmetric space

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

the mean curvature flow for isoparametric submanifolds a submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant we study the mean curvature flow with initial data an isoparametric submanifold in euclidean space and sphere we show that the mean curvature flow preserves the isoparametric condition develops singularities in finite time and converges in finite time to a smooth submanifold of lower dimension we also give a precise description of the collapsing

reducibility of complex submanifolds of the complex euclidean spaces let m be a simply connected complex submanifold of mathbb c n we prove that m is irreducible up a totally geodesic factor if and only if the normal holonomy group acts irreducibly this is an extrinsic analogue of the well known de rham decomposition theorem for a complex manifold our result is not valid in the real context as it is shown by many counter examples

mok s characteristic varieties and the normal holonomy group abstract in this paper we complete the study of the normal holonomy groups of complex submanifolds non nec complete of c n or c p n we show that irreducible but non transitive normal holonomies are exactly the hermitian s representations of 4 table 1 see corollary 1 1 for each one of them we construct a non necessarily complete complex submanifold whose normal holonomy is the prescribed s representation we also show that if the submanifold has irreducible non transitive normal holonomy then it is an open subset of the smooth part of one of the characteristic varieties studied by n mok in his work about rigidity of locally symmetric spaces finally we prove that if the action of the normal holonomy group of a projective submanifold is reducible then the submanifold is an open subset of the smooth part of a so called join i e the union of the lines joining two projective submanifolds

convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems this paper studies the convergence and properties of the solutions of the riccati difference equation special emphasis is given to systems which are not necessarily stabilizable in the filtering sense particularly those having uncontrollable roots on the unit circle besides generalizing and unifying previous work the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components

proper fredholm submanifolds of hilbert space on considere des varietes et des sous varietes de hilbert riemanniennes on demontre qu une sous variete de fredholm propre a une structure de fredholm naturelle et on montre que les operateurs de forme sont compacts et que chaque fonction distance euclidienne fa satisfait la condition c de palais et smale on etudie la geometrie des sous varietes de fredholm propre a fibre normal plat

homogeneity of infinite dimensional isoparametric submanifolds a subsets of a riemannian manifoldn is called extrinsically homogeneous ifs is an orbit of a subgroup of the isometry group ofn in th thorbergsson proved the remarkable result that every complete connected full irreducible isoparametric submanifold of a flnite dimensional euclidean space of rank at least 3 is extrinsically homogeneous this result combined with results of pt1 and da flnally classifled irreducible isoparametric submanifolds of a flnite dimensional euclidean space of rank at least 3 while thorbergsson s proof used tits buildings a simpler proof without using tits buildings was given by olmos cf o2 the main purpose of this paper is to extend thorbergsson s result to the inflnite dimensional case the study of inflnite dimensional isoparametric submanifolds of a hilbert space always assumed to be separable was initiated by terng t2 besides its intrinsic interest the theory of inflnite dimensional isoparametric submanifolds is a very useful tool in studying the submanifold geometry of compact symmetric spaces cf tt as well as hl and ew although much progress has been made especially many basic properties of flnite dimensional isoparametric submanifolds having been successfully extended to the inflnite dimensional case cf t2 t5 and hl the classiflcation of inflnite dimensional isoparametric submanifolds is far from being solved to this end the understanding of the homogeneity of inflnite dimensional isoparametric submanifolds will certainly play a very important role as it does in the flnite dimensional case examples of non homogeneous inflnite dimensional isoparametric hypersurfaces have been found by terng and thorbergsson cf tt in this paper we will prove the following theorem which solves a long standing open problem cf t3 t4 tt theorem a let m be a complete connected irreducible isoparametric submanifold in a hilbert space v assume that the set of all the curvature normals of m at some point is not contained in any a ne line then m is extrinsically homogeneous in the hilbert space v remark in this theorem the dimension of m and thus v could be either flnite or inflnite without loss of generality we may assume that m is full i e

parallel submanifolds of complex projective space and their 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 indeed we explain how these submanifolds can be regarded as the unique complex orbits of the projectivized isotropy representation of an irreducible hermitian symmetric space moreover we show how these important submanifolds are related to other areas of mathematics and theoretical physics finally we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space

submanifold geometry in symmetric spaces the classical local invariants of a submanifold in a space form are the first fundamental form the shape operators and the induced normal connection and they determine the submanifold up to ambient isometry one of the main topics in differential geometry is to study the relation between the local invariants and the global geometry and topology of submanifolds many remarkable results have been developed for submanifolds in space forms whose local invariants satisfy certain natural conditions the study of focal points of a submanifold in an arbitrary riemannian manifold arises from the morse theory of the energy functional on the space of paths in the riemannian manifold joining a fixed point to the submanifold the morse index theorem relates the geometry of a submanifold to the topology of this path space the focal structure is intimately related to the local invariants of the submanifold in the case of space forms one can go backwards and reconstruct the local invariants from the focal structure so it is not too surprising that most of the structure theory of submanifolds can be reformulated in terms of their focal structure what is perhaps surprising is a fact that became increasingly evident to the authors from their individual and joint research over the past decade while extending the theory of submanifolds to ambient spaces more general than space forms proves quite difficult if one tries to use the same approach as for the space forms at least for symmetric spaces it has proved possible to develop an elegant theory based on focal structure that reduces to the classical theory in the case of space forms this paper is an extended report on this theory and the authors believe that the methods developed herein provide important tools for a continuing study of the submanifold geometry in symmetric spaces first we set up some notations let n g be a riemannian manifold m an immersed submanifold of n and m the normal bundle of m the end point map m n of m is the restriction of the exponential map exp to m if v m x is a singular point of and the dimension of the kernel of d v is m then v is called a multiplicity m focal normal and exp v is called a multiplicity m focal point of m with respect to m in n the focal data m is defined to be the set of all pairs v m such that v is a multiplicity m focal normal of m the focal variety v m is the set of all pairs v m with v m m the main purpose of our paper is to study the global geometry and topology

ancient solutions to mean curvature flow for isoparametric submanifolds mean curvature flow for isoparametric submanifolds in euclidean spaces and spheres was studied in liu and terng duke math j 147 1 157 179 2009 in this paper we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity we also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the chern s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres

subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras let a t be the subalgebra of the steenrod algebra generated by sqi i 2t we determine which a module indecomposable summands of f2 x1 xn are free over a t by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of mn f2 or gln f2 occur as composition factors in the truncated polynomial algebra f2 x1 xn x12 xn2 the second question is then answered

parallel focal structure and singular riemannian foliations we give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds we show that this is a singular riemannian foliation with complete orthogonal transversals for this object we construct an action on the transversals that generalizes the weyl group action for polar actions

on the nullity distribution of a submanifold of a space form if m is a submanifold of a space form the nullity distribution mathcal n of its second fundamental form is when defined the common kernel of its shape operators in this paper we will give a local description of any submanifold of the euclidean space by means of its nullity distribution we will also show the following global result if m is a complete irreducible submanifold of the euclidean space or the sphere then its complementary distribution mathcal n bot is completely non integrable this means that any two points in m can be joined by a curve everywhere perpendicular to mathcal n we will finally show that this statement is false for a submanifold of the hyperbolic space

tautness and lie sphere geometry an immersion f of a smooth compact connected manifold m into euclidean space e is said to be taut if every morse function of the form lp x ip f x l 2 p lr has the minimum number of critical points required by the morse inequalities on m carter and west 3 showed that tautness is invariant under moebius conformal transformations of ir and under stereographic projection from ir into the unit sphere s in ir 1 they showed further that a taut immersion must in fact be an embedding pinkall 13 showed that for the theory of taut embeddings it is sufficient to study hypersurfaces in the following sense let m 11 or s be a compact submanifold of codimension greater than one and let m s be a tube of sufficiently small radius about m so that m is an embedded hypersurface then m is taut if and only if m s is taut cecil and ryan 6 showed that any isoparametric hypersurface in s is taut and recently hsiang palais and terng 1 8 have proven that any isoparametric submanifold of any codimension is taut the main result of this paper is that within the class of immersions tautness is invariant under the group of lie sphere transformations this group is considerably larger than the group of moebius transformations as we will now explain in his work on contact transformations lie 1 9 developed his geometry of oriented spheres see also blaschke 2 lie established a bijective correspondence between the set of all oriented hyperspheres and point spheres in s and the points on the quadric hypersurface q i in real projective space pn 2 given by the equation x x 0 where is an indefinite metric with signature n 1 2 on n 3 q contains projective lines but no linear subspaces of pn 2 of higher dimension the 1 parameter family of oriented spheres called a parabolic pencil corresponding to the points of a projective line lying on q l consists of all oriented hyperspheres which are in oriented contact at a certain contact element on s in this way lie established a diffeomorphism between the manifold of contact elements on s and the space a 2 1 of projective lines which lie on q 1 a

isoparametric submanifolds of hyperbolic spaces in this paper we prove a decomposition theorem for isoparametric submanifolds of hyperbolic spaces and as a consequence we obtain all polar actions on hyperbolic spaces we also prove that any isoparametric submanifold of infinite dimensional hyperbolic space is either totally geodesic or finite dimensional

suboptimal control of linear discrete stochastic systems with linear input constraints optimal control of linear discrete stochastic systems with linear input constraints is considered a lower bound on the achievable minimum of the loss function is given a suboptimal control strategy is derived by using a truncated taylor series to approximate the expected future loss in the bellman equation the performance of the suboptimal controller is studied using monte carlo simulation and the obtained loss is compared with the lower bound

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

the topology of isoparametric submanifolds it has been known since a famous paper of bott and samelson that using morse theory the homology and cohomology of certain homogeneous spaces can be computed algorithmically from dynkin diagram and multiplicity data l conlon and j dadok noted that these spaces are the orbits of the isotropy representations of symmetric spaces recently the theory of isoparametric hypersurfaces has been generalized to a theory of isoparametric submanifolds of arbitrary codimension in euclidean space and these same orbits turn out to be exactly the homogeneous examples even the nonhomogeneous examples have associated to them weyl groups with dynkin diagrams marked with multiplicities we extend and simplify the bott samelson method to compute the homology and cohomology of isoparametric submanifolds from their marked dynkin diagrams

discrete minimax linear quadratic regulation of continuous time systems abstract in this paper the statistical cyclostationary characteristics of a controlled continuous time system which result as a consequence of implementing a digital controller are investigated a minimax quadratic cost function which takes into account the periodic nature of the statistics is defined and optimized the resulting linear state feedback law improves the intersample behavior of the controlled system when compared to discrete time optimization in particular the minimax variance regulator which is proposed as a new design method for regulating the output variance offers a significant improvement over the classical minimum variance regulator

dupin hypersurfaces with three principal curvatures using the method of moving frames we prove that any locally irreducible dupin hypersurface in sn with three distinct principal curvatures is equivalent by lie sphere transformation to an isoparametric hypersurface in sn

a berger type normal holonomy theorem for complex submanifolds we prove a berger type theorem for the normal holonomy phi perp i e the holonomy group of the normal connection of a full complete complex submanifold m of the complex projective space mathbb c p n namely if phi perp does not act transitively then m is the complex orbit in the complex projective space of the isotropy representation of an irreducible hermitian symmetric space of rank greater or equal to 3 moreover we show that for complete irreducible complex submanifolds of mathbb c n the normal holonomy is generic i e it acts transitively on the unit sphere of the normal space the methods in the proofs rely heavily on the singular data of appropriate holonomy tubes after lifting the submanifold to the complex euclidean space in the mathbb c p n case and basic facts of complex submanifolds

decomposing thick subcategories of the stable module category let underline rm mod kg be the stable category of finitely generated modular representations of a finite group g over a field k we prove a krull remak schmidt theorem for thick subcategories of underline rm mod kg it is shown that every thick tensor ideal mathcal c of underline rm mod kg i e a thick subcategory which is a tensor ideal has a usually infinite unique decomposition mathcal c coprod i in i mathcal c i into indecomposable thick tensor ideals this decomposition follows from a decomposition of the corresponding idempotent kg module e mathcal c into indecomposable modules if mathcal c mathcal c w is the thick tensor ideal corresponding to a closed homogeneous subvariety w of the maximal ideal spectrum of the cohomology ring h g k then the decomposition of mathcal c reflects the decomposition w bigcup i 1 nw i of w into connected components

the normal holonomy group of kahler submanifolds we study the restricted holonomy group rm hol nabla perp of the normal connection nabla perp shortened to normal holonomy group of a kahler submanifold of a complex space form we prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible hermitian symmetric space in particular it is a compact group and the complex structure j belongs to its lie algebra we prove that the normal holonomy group acts irreducibly if the submanifold is full that is it is not contained in a totally geodesic proper kahler submanifold and the second fundamental form at some point has no kernel for example a kahler einstein submanifold of mathbb c p n has this property we define a new invariant mu of a kahler submanifold of a complex space form for non full submanifolds the invariant mu measures the deviation of j from belonging to the normal holonomy algebra for a kahler einstein submanifold the invariant mu is a rational function of the einstein constant by using the invariant mu we prove that the normal holonomy group of a not necessarily full kahler einstein submanifold of mathbb c p n is compact and we give a list of possible holonomy groups the approach is based on a definition of the holonomy algebra rm hol p of an arbitrary curvature tensor field p on a vector bundle with a connection and on a de rham type decomposition theorem for rm hol p

isoparametric hypersurfaces with g m 6 2 we prove that isoparametric hypersurfaces with g m 6 2 are homogeneous which answers dorfmeister neher s conjecture armatively and solves yau s problem in the case g 6

submanifolds with curvature normals of constant length and the gauss map we show that a submanifold with curvature normal of constant length has constant principal curvatures under suitable global hypothesis we construct local counterexamples to show that the global hypothesis can not be dropped

representations of the double burnside algebra and cohomology of the extraspecial p group ii abstract let e be the extraspecial p group of order p 3 and exponent p where p is an odd prime we determine the mod p cohomology h x f p of a summand x in the stable splitting of p completed classifying space be in the previous paper hida and yagita 2014 7 we determined this cohomology modulo nilpotence in this paper we consider the whole of the cohomology moreover we consider the stable splittings of bg for some finite groups with sylow p subgroup e related to the three dimensional linear group l 3 p

isoparametric hypersurfaces with four principal curvatures let m be an isoparametric hypersurface in the sphere s n with four distinct principal curvatures munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 m 2 and stolz showed that the pair m1 m 2 must either be 2 2 4 5 or be equal to the multiplicities of an isoparametric hypersurface of fkm type constructed by ferus karcher and munzner from orthogonal representations of clifford algebras in this paper we prove that if the multiplicities satisfy m2 2m1 1 then the isoparametric hypersurface m must be of fkm type together with known results of takagi for the case m1 1 and ozeki and takeuchi for m1 2 this handles all possible pairs of multiplicities except for four cases for which the classification problem remains open

mobius curvature laguerre curvature and dupin hypersurface abstract in this paper we show that a dupin hypersurface with constant mobius curvatures is mobius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere we also show that a dupin hypersurface with constant laguerre curvatures is laguerre equivalent to a flat laguerre isoparametric hypersurface these results solve the major issues related to the conjectures of cecil et al on the classification of dupin hypersurfaces

a geometric proof of the berger holonomy theorem we give a geometric proof of the berger holonomy theorem the proof uses euclidean submanifold geometry of orbits and gives a link between riemannian holonomy groups and normal holonomy groups

isoparametric foliations diffeomorphism groups and exotic smooth structures abstract in this paper we are concerned with interactions between isoparametric theory and differential topology two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves using differential topology we obtain several results towards the classification problem of isoparametric foliations up to equivalence in particular we show that each homotopy n sphere has the same isoparametric foliations as the standard sphere s n has except for n 4 reducing the classification problem on homotopy spheres to that on the standard sphere moreover we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere s n except for n 5 besides we show that the uniqueness holds on s 5 if and only if 0 diff s 4 z 2 i e pseudo isotopy implies isotopy for diffeomorphisms on s 4 at last some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds

polar orthogonal representations of real reductive algebraic groups we prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo riemannian symmetric space we also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive lie algebras

multiplicities of dupin hypersurfaces all known examples of isoparametric hypersurfaces of spheres are of the following two types 1 the homogeneous examples which are principal orbits of isotropy representations of symmetric spaces of rank 2 2 the clifford examples of ferus karcher munzner which are constructed from modules over clifford algebras our main result which extends work of munzner thorbergsson and grove halperin is that any isoparametric hypersurface has the same number of principal curvatures and the same multiplicities as some hypersurface of type 1 or 2 more generally this holds for any dupin hypersurface the proof is homotopy theoretic the main tools are the hopf invariant and the ehp sequence

geometry and topology of isoparametric submanifolds in euclidean spaces abstract restrictions are given on the possible marked dynkin diagrams of isoparametric submanifolds and their homology and cohomology are computed by an extension of the techniques used by bott and samelson bott r samelson h 1958 am j math 80 964 1029 and by borel borel a 1953 ann math 57 2 115 207 for g t

a general theory of canonical forms si g est un groupe de lie compact et m une g variete riemannienne d orbites principales de codimension k alors une section ou une forme canonique pour m est une sous variete fermee lisse a k dimensions de m qui rencontre toutes les orbites de m orthogonalement