1 First and second variational formulas for area
|
|
- Scarlett Oliver
- 5 years ago
- Views:
Transcription
1 1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on the volume and Laplacian comparison theorems. It will also be used in our studies of the stability issues of minimal submanifolds. Let M be a Riemannian manifold of dimension m with metric denoted by ds 2. In terms of local coordinates {x 1,...,x m } the metric is written in the form ds 2 g ij dx i dx j, where we are adopting the convention that repeated indices are being summed over. If X and Y are tangent vectors at a point p M, we will also denote their inner product by ds 2 (X, Y ) X, Y. If we let S(TM) be the set of smooth vector fields on M, then the Riemannian connection :S(TM) S(TM) S(TM) satisfies the following properties: (1) ( f1 X 1 + f 2 X 2 )Y f 1 X1 Y + f 2 X2 Y ; (2) X ( f 1 Y 1 + f 2 Y 2 ) X ( f 1 )Y 1 + f 1 X Y 1 + X ( f 2 )Y 2 + f 2 X Y 2 ; (3) XY 1, Y 2 X Y 1, Y 2 +Y 1, X Y 2 ; and (4) X Y Y X [X, Y ], for all X, Y S(TM), for all X, X 1, X 2, Y, Y 1, Y 2 S(TM) and for all f 1, f 2 C (M). Property (3) says that is compatible with the Riemannian metric, while property (4) means that is torsion free. Moreover, the Riemannian connection is the 1 in this web service
2 2 Geometric Analysis only connection satisfying the above properties. The curvature tensor of the Riemannian metric is then given by R XY Z X Y Z Y X Z [X,Y ] Z, for X, Y, Z S(TM), and it satisfies the properties: (1) R XY Z R YX Z; (2) R XY Z + R YZ X + R ZX Y ; and (3) R XY Z, W R ZW X, Y, for all X, Y, Z, W S(TM). The sectional curvature of the 2-plane section spanned by a pair of orthonormal vectors X and Y is defined by K (X, Y ) R XY Y, X. If we take {e 1,...,e m } to be an orthonormal basis of the tangent space of M, then the Ricci curvature is defined to be the symmetric 2-tensor given by R ij m R ei,e k e k, e j. k1 Observe that the diagonal elements of the Ricci curvature are given by R ii k i K (e i, e k ). Let be an n-dimensional submanifold in M with n < m. The Riemannian metric dsm 2 defined on M when restricted to induces a Riemannian metric ds 2 on. One can easily check that for vector fields X, Y S(T), if we define t X Y ( X Y ) t to be the tangential component of X Y to, then t is the Riemannian connection of with respect to ds 2. The normal component of yields the negative of the second fundamental form of. In particular, one defines the second fundamental form by II(X, Y ) ( X Y ) n, and checks that it is tensorial with respect to X, Y S(T ). Taking the trace of the bilinear form II over the tangent space of yields the mean curvature vector, given by tr( II) H. in this web service
3 1 First and second variational formulas for area 3 Let us now consider a one-parameter family of deformations of given by t φ(, t) for t ( ɛ, ɛ) with. Let {x 1,...,x n } be a coordinate system around a point p. We can consider {x 1,...,x n, t} to be a coordinate system of ( ɛ, ɛ) near the point (p, ). Let us denote e i dφ ( / x i ) for i 1,...,n and T dφ ( / t). The induced metric on t from M is then given by g ij e i, e j. We may further assume that {x 1,...,x n } form a normal coordinate system at p. Hence g ij (p, ) δ ij and ei e j (p, ). Let us define da t to be the area element of t with respect to the induced metric. For t sufficiently close to, we can write da t J(x, t) da. With respect to the normal coordinate system {x 1,...,x n }, the function J(x, t) is given by J(x, t) g(x, t) g(x, ) with g(x, t) det(g ij (x, t)). To compute the first variation for the area of, we compute J (p, t) ( J/ t)(p, t). By the assumption that g ij (p, ) δ ij, we have J (p, ) 1 2 g (p, ). However, g det(g ij ) g 1 j c 1 j, j1 where c ij are the cofactors of g ij. Therefore g (p, ) g 1 j (p, ) c 1 j (p, ) + j1 g 11 (p, ) + c 11 (p, ). j1 g 1 j (p, ) c 1 j (p, ) By induction on the dimension, we conclude that g (p, ) n i 1 g ii. On the other hand, g ii T e i, e i 2 T e i, e i 2 ei T, e i, because {x 1,...,x n, t} form a coordinate system for ( ɛ, ɛ). Let us point out that the quantity ei T, e i i1 in this web service
4 4 Geometric Analysis is now well defined under an orthonormal change of basis and hence is globally defined. If we write T T t + T n, where T t is the tangential component of T on and T n is its normal component, then ei T, e i i1 ei T t, e i + i1 div(t t ) + ei T n, e i i1 e i T n, e i i1 div(t t ) +T n, H, T n, ei e i where H is the mean curvature vector of. Hence the first variation for the volume form at the point (p, ) is given by d ( dt da t (p,) divt t + T n, ) (p,) H da. However, the right-hand side is intrinsically defined independent of the choice of coordinates and hence this formula is valid at any arbitrary point. If T is a compactly supported variational vector field on, then using the divergence theorem the first variation of the area of is given by d dt A( t) t T n, H This shows that the mean curvature of is identically if and only if is a critical point of the area functional. Definition 1.1 An immersed submanifold M is said to be minimal if its mean curvature vector vanishes identically, i.e., H. When is a curve in M that is parametrized by arc-length with unit tangent vector e, then the first variational formula for length can be written as d dt L T t, e l l T n, e e t. i1 l T, e l T, e e. We will now proceed to derive the second variational formula for area. Let φ : ( ɛ, ɛ) ( ɛ, ɛ) M be a two-parameter family of variations of. Using similar notation, we write dφ( / x i ) e i for i 1,...,n, and denote the variational vector fields by dφ( / t) T and dφ( / s) S. in this web service
5 1 First and second variational formulas for area 5 In terms of a general coordinate system, the first partial derivative of J can be written as J (x, t, s) t i, j1 g ij ei T, e j J(x, t, s), where (g ij ) denotes the inverse matrix of (g ij ). Differentiating this with respect to s and evaluating at (p,, ) we have 2 J s t i, j1 ( ) S g ij ei T, e j J (Sg ij ) ei T, e j J + i, j1 + g ij ei T, e j S(J) i, j1 (Sg ij ) ei T, e j + i, j1 i1 g ij ( S ei T, e j ) J i, j1 S ei T, e i i1 ( + ei T, e i ) e j S, e j. (1.1) However, differentiating the formula n k 1 g ik g kj δ ij, we obtain j1 hence (Sg ik )g kj g ik (Sg kj ), k1 k1 Sg ij g ik (Sg kl )g lj k,l1 Sg ij Se i, e j S e i, e j S e j, e i ei S, e j e j S, e i. in this web service
6 6 Geometric Analysis The first term on the right-hand side of (1.1) now becomes (Sg ij ) ei T, e j i, j1 ei S, e j ei T, e j i, j1 e j S, e i ei T, e j. i, j1 The second term on the right-hand side of (1.1) can be written as S ei T, e i i1 S ei T, e i + i1 R Sei T, e i + i1 ei T, S e i i1 ei S T, e i + i1 ei T, ei S, where the term R Sei T, e i on the right-hand side denotes the curvature tensor of M. Therefore, we have 2 J s t + ei S, e j ei T, e j i, j1 R Sei T, e i + i1 i1 i1 e j S, e i ei T, e j i, j1 ei S T, e i + i1 ei T, ei S ( + ei T, e i ) e j S, e j. (1.2) j1 We will now consider some special cases that will simplify (1.2). Let us first assume that is a curve parametrized by arc-length in M with unit tangent vector given by e, then the second variational formula for the length is given by 2 L s t (s,t)(,) l + l i1 { e S, e e T, e +R Se T, e } { e S T, e + e T, e S }. in this web service
7 1 First and second variational formulas for area 7 If we further assumed that is a geodesic satisfying the geodesic equation e e, then we have 2 L l s t { (es, e )(et, e ) +R Se T, e } (s,t)(,) l + {e S T, e + e T, e S } l { e T, e S +R Se T, e (es, e )(et, e ) } + S T, e l. The second special case is when is a general n-dimensional manifold and then if the two variational vector fields are the same and are normal to, (1.2) becomes 2 J t 2 t + ei T, e j 2 i, j1 ei T T, e i + i1 ei T, e j 2 i, j1 e j T, e i ei T, e j + i, j1 R Tei T, e i i1 ( ei T 2 + ei T, e i i1 i1 e j T, e i ei T, e j i, j1 + div( T T ) t + ( T T ) n, H + i1 ) 2 R ei T T, e i i1 ei T 2 + T, 2 H. (1.3) On the other hand, if {e n+1,...,e m } denotes an orthonormal set of vectors normal to in M, then ei T, ei T i1 ei T, e j 2 + i, j1 m i1 νn+1 ei T, e ν 2. Also ei T, e j T, II ij e j T, e i, in this web service
8 8 Geometric Analysis where II ij denotes the second fundamental form with value in the normal bundle of. Hence, (1.3) becomes 2 J t 2 T, 2 II ij R ei T T, e i +div( T T ) t t i, j d 2 dt 2 A( t) t + ( T T ) n, H + i1 m i1 νn+1 ei T, e ν 2 + T, 2 H. Therefore, the second variational formula for area in terms of compactly supported normal variations is given by T, 2 II ij i, j + m i1 νn+1 i1 R ei T T, e i + ei T, e ν 2 + T, 2 H. ( T T ) n, H Definition 1.2 A minimally immersed submanifold M is said to be stable if the second variation for area with respect to all compactly supported normal variations is nonnegative. This means that the stability inequality T, 2 m II ij R ei T T, e i + ei T, e ν 2 i, j i1 i1 νn+1 is valid for any compactly supported normal vector field T. If we further restrict to be an orientable codimension-1 minimal submanifold of an orientable manifold M, we can write any normal variation in the form T ψe m, where ψ is a differentiable function on and e m is a unit normal vector field to. Then the second variational formula can be written as d 2 dt 2 A( t) t T, 2 II ij R(T, T ) + ei T, e m 2 i, j i1 { ψ 2 h 2 ij ψ2 R(e m, e m ) + ψ 2}, where II ij h ij e m with h ij being the component of the second fundamental form and R(T, T ) denotes the Ricci curvature of M in the direction of T. Here we have also used the fact that in this web service
9 1 First and second variational formulas for area 9 ei T, e m ψ ei e m, e m +e i (ψ)e m, e m e i (ψ). In particular, the stability inequality in this case is given by ψ 2 ψ 2 hij 2 + ψ 2 R(e m, e m ). (1.4) The last special case is again to assume that is an oriented hypersurface in an oriented manifold M and we restrict the variation to be given by hypersurfaces which are a constant distant from. The variational vector field is then given by e m with em e m. This situation is particularly useful for the purpose of controlling the growth of the volume of geodesic balls of radius r. In this case, if we write H He m, the first variational formula for the area element becomes J (x, ) H(x) J(x, ), (1.5) t and the second variational formula can be written as 2 m 1 J t 2 (x, ) i, j1 hij 2 (x) J(x, ) R(e m, e m )(x) J(x, ) + H 2 (x) J(x, ). (1.6) in this web service
10 2 Volume comparison theorem In this chapter, we will develop a volume comparison theorem originally proved by Bishop (see [BC]). Let p M be a point in a complete Riemannian manifold of dimension m. In terms of polar normal coordinates at p, we can write the volume element as J(θ, r)dr dθ, where dθ is the area element of the unit (m 1)-sphere. The Gauss lemma asserts that the area element of submanifold B p (r), which is the boundary of the geodesic ball of radius r, is given by J(θ, r)dθ. By the first and second variational formulas (1.5) and (1.6), ifx (θ, r) is not in the cut-locus of p, we have and J (θ, r) 2 J (θ, r) r 2 m 1 i, j1 J (θ, r) J (θ, r) r H(θ, r) J(θ, r) (2.1) h 2 ij (θ, r) J(θ, r) R rr(θ, r) J(θ, r) + H 2 (θ, r) J(θ, r), (2.2) where R rr R( / r, / r), H(θ, r), and (h ij (θ, r)) denote the Ricci curvature in the radial direction, the mean curvature and the second fundamental form of B p (r) at the point x (θ, r) with respect to the unit normal vector / r, respectively. 1 in this web service
DIFFERENTIAL GEOMETRY HW 12
DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationSection 6. Laplacian, volume and Hessian comparison theorems
Section 6. Laplacian, volume and Hessian comparison theorems Weimin Sheng December 27, 2009 Two fundamental results in Riemannian geometry are the Laplacian and Hessian comparison theorems for the distance
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More informationWarped Products. by Peter Petersen. We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion
Warped Products by Peter Petersen De nitions We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion a function the di erential v = dx i (v) df = f dxi We
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationRICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS
J. Austral. Math. Soc. 72 (2002), 27 256 RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS ION MIHAI (Received 5 June 2000; revised 19 February 2001) Communicated by K. Wysocki Abstract Recently,
More informationWARPED PRODUCTS PETER PETERSEN
WARPED PRODUCTS PETER PETERSEN. Definitions We shall define as few concepts as possible. A tangent vector always has the local coordinate expansion v dx i (v) and a function the differential df f dxi We
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationSection 2. Basic formulas and identities in Riemannian geometry
Section 2. Basic formulas and identities in Riemannian geometry Weimin Sheng and 1. Bianchi identities The first and second Bianchi identities are R ijkl + R iklj + R iljk = 0 R ijkl,m + R ijlm,k + R ijmk,l
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationHomework for Math , Spring 2012
Homework for Math 6170 1, Spring 2012 Andres Treibergs, Instructor April 24, 2012 Our main text this semester is Isaac Chavel, Riemannian Geometry: A Modern Introduction, 2nd. ed., Cambridge, 2006. Please
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More information5.2 The Levi-Civita Connection on Surfaces. 1 Parallel transport of vector fields on a surface M
5.2 The Levi-Civita Connection on Surfaces In this section, we define the parallel transport of vector fields on a surface M, and then we introduce the concept of the Levi-Civita connection, which is also
More informationLet F be a foliation of dimension p and codimension q on a smooth manifold of dimension n.
Trends in Mathematics Information Center for Mathematical Sciences Volume 5, Number 2,December 2002, Pages 59 64 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS KYOUNG HEE HAN AND HOBUM KIM Abstract.
More informationPHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH
PHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH 1. Differential Forms To start our discussion, we will define a special class of type (0,r) tensors: Definition 1.1. A differential form of order
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More informationHOMEWORK 2 - RIEMANNIAN GEOMETRY. 1. Problems In what follows (M, g) will always denote a Riemannian manifold with a Levi-Civita connection.
HOMEWORK 2 - RIEMANNIAN GEOMETRY ANDRÉ NEVES 1. Problems In what follows (M, g will always denote a Riemannian manifold with a Levi-Civita connection. 1 Let X, Y, Z be vector fields on M so that X(p Z(p
More informationCHAPTER 1 PRELIMINARIES
CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable
More informationarxiv: v1 [math.dg] 21 Sep 2007
ON THE GAUSS MAP WITH VANISHING BIHARMONIC STRESS-ENERGY TENSOR arxiv:0709.3355v1 [math.dg] 21 Sep 2007 WEI ZHANG Abstract. We study the biharmonic stress-energy tensor S 2 of Gauss map. Adding few assumptions,
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationDifferential Geometry of Surfaces
Differential Forms Dr. Gye-Seon Lee Differential Geometry of Surfaces Philipp Arras and Ingolf Bischer January 22, 2015 This article is based on [Car94, pp. 77-96]. 1 The Structure Equations of R n Definition
More informationON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES
ON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES CLAUS GERHARDT Abstract. We prove that the mean curvature τ of the slices given by a constant mean curvature foliation can be used
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More information1 Normal (geodesic) coordinates
Riemannian Geometry The Bochner- Weitzenbock formula If we need to verify some tensor identity (or inequality) on Riemannina manifolds, we only need to choose, at every point, a suitable local coordinate,
More informationDIFFERENTIAL GEOMETRY. LECTURE 12-13,
DIFFERENTIAL GEOMETRY. LECTURE 12-13, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of
More informationConstant mean curvature biharmonic surfaces
Constant mean curvature biharmonic surfaces Dorel Fetcu Gheorghe Asachi Technical University of Iaşi, Romania Brest, France, May 2017 Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May 2017 1 / 21
More informationA brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationTHE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS
THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationOn Null 2-Type Submanifolds of the Pseudo Euclidean Space E 5 t
International Mathematical Forum, 3, 2008, no. 3, 609-622 On Null 2-Type Submanifolds of the Pseudo Euclidean Space E 5 t Güler Gürpınar Arsan, Elif Özkara Canfes and Uǧur Dursun Istanbul Technical University,
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationAn Inequality for Warped Product Semi-Invariant Submanifolds of a Normal Paracontact Metric Manifold
Filomat 31:19 (2017), 6233 620 https://doi.org/10.2298/fil1719233a Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat An Inequality for
More information1 Curvature of submanifolds of Euclidean space
Curvature of submanifolds of Euclidean space by Min Ru, University of Houston 1 Curvature of submanifolds of Euclidean space Submanifold in R N : A C k submanifold M of dimension n in R N means that for
More informationMathematical Relativity, Spring 2017/18 Instituto Superior Técnico
Mathematical Relativity, Spring 2017/18 Instituto Superior Técnico 1. Starting from R αβµν Z ν = 2 [α β] Z µ, deduce the components of the Riemann curvature tensor in terms of the Christoffel symbols.
More informationMATH DIFFERENTIAL GEOMETRY. Contents
MATH 3968 - DIFFERENTIAL GEOMETRY ANDREW TULLOCH Contents 1. Curves in R N 2 2. General Analysis 2 3. Surfaces in R 3 3 3.1. The Gauss Bonnet Theorem 8 4. Abstract Manifolds 9 1 MATH 3968 - DIFFERENTIAL
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More informationJeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi
Bull. Korean Math. Soc. 40 (003), No. 3, pp. 411 43 B.-Y. CHEN INEQUALITIES FOR SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Abstract. Some B.-Y.
More informationOn constant isotropic submanifold by generalized null cubic
On constant isotropic submanifold by generalized null cubic Leyla Onat Abstract. In this paper we shall be concerned with curves in an Lorentzian submanifold M 1, and give a characterization of each constant
More informationCENTROAFFINE HYPEROVALOIDS WITH EINSTEIN METRIC
CENTROAFFINE HYPEROVALOIDS WITH EINSTEIN METRIC Udo Simon November 2015, Granada We study centroaffine hyperovaloids with centroaffine Einstein metric. We prove: the hyperovaloid must be a hyperellipsoid..
More informationON THE LOCAL VERSION OF THE CHERN CONJECTURE: CMC HYPERSURFACES WITH CONSTANT SCALAR CURVATURE IN S n+1
Kragujevac Journal of Mathematics Volume 441 00, Pages 101 111. ON THE LOCAL VERSION OF THE CHERN CONJECTURE: CMC HYPERSURFACES WITH CONSTANT SCALAR CURVATURE IN S n+1 S. C. DE ALMEIDA 1, F. G. B. BRITO,
More informationRiemannian Manifolds
Riemannian Manifolds Joost van Geffen Supervisor: Gil Cavalcanti June 12, 2017 Bachelor thesis for Mathematics and Physics Contents 1 Introduction 2 2 Riemannian manifolds 4 2.1 Preliminary Definitions.....................................
More informationH-convex Riemannian submanifolds
H-convex Riemannian submanifolds Constantin Udrişte and Teodor Oprea Abstract. Having in mind the well known model of Euclidean convex hypersurfaces [4], [5] and the ideas in [1], many authors defined
More informationLECTURE 8: THE SECTIONAL AND RICCI CURVATURES
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to
More informationGENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE. 1. Introduction
ACTA MATHEMATICA VIETNAMICA 205 Volume 29, Number 2, 2004, pp. 205-216 GENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE HANDAN BALGETIR AND MAHMUT ERGÜT Abstract. In this paper, we define
More informationÜbungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.
Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση
More informationLECTURE 24: THE BISHOP-GROMOV VOLUME COMPARISON THEOREM AND ITS APPLICATIONS
LECTURE 24: THE BISHOP-GROMOV VOLUME COMPARISON THEOREM AND ITS APPLICATIONS 1. The Bishop-Gromov Volume Comparison Theorem Recall that the Riemannian volume density is defined, in an open chart, to be
More informationIntroduction to Differential Geometry
More about Introduction to Differential Geometry Lecture 7 of 10: Dominic Joyce, Oxford University October 2018 EPSRC CDT in Partial Differential Equations foundation module. These slides available at
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationLECTURE 10: THE PARALLEL TRANSPORT
LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be
More information2-harmonic maps and their first and second variational formulas i
Note di atematica Note at. 1(2008, suppl. n. 1, 209-232 ISSN 1123-2536, e-issn 1590-0932 DOI 10.1285/i15900932v28n1supplp209 Note http://siba2.unile.it/notemat di atematica 28, suppl. n. 1, 2009, 209 232.
More information4.7 The Levi-Civita connection and parallel transport
Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves
More informationGENERALIZED ISOPERIMETRIC INEQUALITIES FOR EXTRINSIC BALLS IN MINIMAL SUBMANIFOLDS. Steen Markvorsen and Vicente Palmer*
GENEALIZED ISOPEIMETIC INEQUALITIES FO EXTINSIC BALLS IN MINIMAL SUBMANIFOLDS Steen Markvorsen and Vicente Palmer* Abstract. The volume of an extrinsic ball in a minimal submanifold has a well defined
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationA CHARACTERIZATION OF WARPED PRODUCT PSEUDO-SLANT SUBMANIFOLDS IN NEARLY COSYMPLECTIC MANIFOLDS
Journal of Mathematical Sciences: Advances and Applications Volume 46, 017, Pages 1-15 Available at http://scientificadvances.co.in DOI: http://dx.doi.org/10.1864/jmsaa_71001188 A CHARACTERIATION OF WARPED
More informationThe spectral action for Dirac operators with torsion
The spectral action for Dirac operators with torsion Christoph A. Stephan joint work with Florian Hanisch & Frank Pfäffle Institut für athematik Universität Potsdam Tours, ai 2011 1 Torsion Geometry, Einstein-Cartan-Theory
More informationLegendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator
Note di Matematica 22, n. 1, 2003, 9 58. Legendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator Tooru Sasahara Department of Mathematics, Hokkaido University, Sapporo 060-0810,
More informationSurfaces with Parallel Mean Curvature in S 3 R and H 3 R
Michigan Math. J. 6 (202), 75 729 Surfaces with Parallel Mean Curvature in S 3 R and H 3 R Dorel Fetcu & Harold Rosenberg. Introduction In 968, J. Simons discovered a fundamental formula for the Laplacian
More informationThe Hodge Star Operator
The Hodge Star Operator Rich Schwartz April 22, 2015 1 Basic Definitions We ll start out by defining the Hodge star operator as a map from k (R n ) to n k (R n ). Here k (R n ) denotes the vector space
More informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, February 25, 1997 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, February 25, 1997 (Day 1) 1. Factor the polynomial x 3 x + 1 and find the Galois group of its splitting field if the ground
More informationDifferential Geometry Exercises
Differential Geometry Exercises Isaac Chavel Spring 2006 Jordan curve theorem We think of a regular C 2 simply closed path in the plane as a C 2 imbedding of the circle ω : S 1 R 2. Theorem. Given the
More informationSyllabus. May 3, Special relativity 1. 2 Differential geometry 3
Syllabus May 3, 2017 Contents 1 Special relativity 1 2 Differential geometry 3 3 General Relativity 13 3.1 Physical Principles.......................................... 13 3.2 Einstein s Equation..........................................
More informationCoordinate Finite Type Rotational Surfaces in Euclidean Spaces
Filomat 28:10 (2014), 2131 2140 DOI 10.2298/FIL1410131B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Coordinate Finite Type
More informationHyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
More informationInformation geometry for bivariate distribution control
Information geometry for bivariate distribution control C.T.J.Dodson + Hong Wang Mathematics + Control Systems Centre, University of Manchester Institute of Science and Technology Optimal control of stochastic
More informationStable minimal cones in R 8 and R 9 with constant scalar curvature
Revista Colombiana de Matemáticas Volumen 6 (2002), páginas 97 106 Stable minimal cones in R 8 and R 9 with constant scalar curvature Oscar Perdomo* Universidad del Valle, Cali, COLOMBIA Abstract. In this
More informationarxiv: v1 [math.dg] 20 Dec 2016
LAGRANGIAN L-STABILITY OF LAGRANGIAN TRANSLATING SOLITONS arxiv:161.06815v1 [math.dg] 0 Dec 016 JUN SUN Abstract. In this paper, we prove that any Lagrangian translating soliton is Lagrangian L-stable.
More informationDifferential Geometry II Lecture 1: Introduction and Motivation
Differential Geometry II Lecture 1: Introduction and Motivation Robert Haslhofer 1 Content of this lecture This course is on Riemannian geometry and the geometry of submanifol. The goal of this first lecture
More informationDifferential Forms, Integration on Manifolds, and Stokes Theorem
Differential Forms, Integration on Manifolds, and Stokes Theorem Matthew D. Brown School of Mathematical and Statistical Sciences Arizona State University Tempe, Arizona 85287 matthewdbrown@asu.edu March
More informationTimelike Rotational Surfaces of Elliptic, Hyperbolic and Parabolic Types in Minkowski Space E 4 with Pointwise 1-Type Gauss Map
Filomat 29:3 (205), 38 392 DOI 0.2298/FIL50338B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Timelike Rotational Surfaces of
More informationChapter 7 Curved Spacetime and General Covariance
Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationVolume comparison theorems without Jacobi fields
Volume comparison theorems without Jacobi fields Dominique Bakry Laboratoire de Statistique et Probabilités Université Paul Sabatier 118 route de Narbonne 31062 Toulouse, FRANCE Zhongmin Qian Mathematical
More information2 Lie Groups. Contents
2 Lie Groups Contents 2.1 Algebraic Properties 25 2.2 Topological Properties 27 2.3 Unification of Algebra and Topology 29 2.4 Unexpected Simplification 31 2.5 Conclusion 31 2.6 Problems 32 Lie groups
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationFundamental Materials of Riemannian Geometry
Chapter 1 Fundamental aterials of Riemannian Geometry 1.1 Introduction In this chapter, we give fundamental materials in Riemannian geometry. In this book, we assume basic materials on manifolds. We give,
More informationCOMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES. 1. Introduction
COMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES ANTONIO CAÑETE AND CÉSAR ROSALES Abstract. We consider a smooth Euclidean solid cone endowed with a smooth
More informationGeometry of Skeletal Structures and Symmetry Sets
Geometry of Skeletal Structures and Symmetry Sets Azeb Zain Jafar Alghanemi Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Pure
More informationThe Laplace-Beltrami-Operator on Riemannian Manifolds. 1 Why do we need the Laplace-Beltrami-Operator?
Frank Schmidt Computer Vision Group - Technische Universität ünchen Abstract This report mainly illustrates a way to compute the Laplace-Beltrami-Operator on a Riemannian anifold and gives information
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationCHARACTERIZATION OF TOTALLY GEODESIC SUBMANIFOLDS IN TERMS OF FRENET CURVES HIROMASA TANABE. Received October 4, 2005; revised October 26, 2005
Scientiae Mathematicae Japonicae Online, e-2005, 557 562 557 CHARACTERIZATION OF TOTALLY GEODESIC SUBMANIFOLDS IN TERMS OF FRENET CURVES HIROMASA TANABE Received October 4, 2005; revised October 26, 2005
More informationHarmonic Morphisms - Basics
Department of Mathematics Faculty of Science Lund University Sigmundur.Gudmundsson@math.lu.se March 11, 2014 Outline Harmonic Maps in Gaussian Geometry 1 Harmonic Maps in Gaussian Geometry Holomorphic
More informationAn introduction to General Relativity and the positive mass theorem
An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More informationA Semi-Riemannian Manifold of Quasi-Constant Curvature Admits Lightlike Submanifolds
International Journal of Mathematical Analysis Vol. 9, 2015, no. 25, 1215-1229 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.5255 A Semi-Riemannian Manifold of Quasi-Constant Curvature
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationWARPED PRODUCT METRICS ON (COMPLEX) HYPERBOLIC MANIFOLDS
WARPED PRODUCT METRICS ON COMPLEX) HYPERBOLIC MANIFOLDS BARRY MINEMYER Abstract. In this paper we study manifolds of the form X \ Y, where X denotes either H n or CH n, and Y is a totally geodesic submanifold
More informationDistances, volumes, and integration
Distances, volumes, and integration Scribe: Aric Bartle 1 Local Shape of a Surface A question that we may ask ourselves is what significance does the second fundamental form play in the geometric characteristics
More informationCS 468, Lecture 11: Covariant Differentiation
CS 468, Lecture 11: Covariant Differentiation Adrian Butscher (scribe: Ben Mildenhall) May 6, 2013 1 Introduction We have talked about various extrinsic and intrinsic properties of surfaces. Extrinsic
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), 79 7 www.emis.de/journals ISSN 176-0091 WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS ADELA MIHAI Abstract. B.Y. Chen
More informationESTIMATES ON THE GRAPHING RADIUS OF SUBMANIFOLDS AND THE INRADIUS OF DOMAINS
ESTIMATES ON THE GRAPHING RADIUS OF SUBMANIFOLDS AND THE INRADIUS OF DOMAINS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. Let
More informationOn some special vector fields
On some special vector fields Iulia Hirică Abstract We introduce the notion of F -distinguished vector fields in a deformation algebra, where F is a (1, 1)-tensor field. The aim of this paper is to study
More informationSome Research Themes of Aristide Sanini. 27 giugno 2008 Politecnico di Torino
Some Research Themes of Aristide Sanini 27 giugno 2008 Politecnico di Torino 1 Research themes: 60!s: projective-differential geometry 70!s: Finsler spaces 70-80!s: geometry of foliations 80-90!s: harmonic
More informationEnergy method for wave equations
Energy method for wave equations Willie Wong Based on commit 5dfb7e5 of 2017-11-06 13:29 Abstract We give an elementary discussion of the energy method (and particularly the vector field method) in the
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More information