International Journal of Research and Review www.ijrrjournal.com E-ISSN: 2349-9788; P-ISSN: 2454-2237
Original Research Article
Analytic Differential Geometry with Manifolds
Mohamed M. Osman
Department of Mathematics. Faculty of Science, University of Al-Baha, Kingdom of Saudi Arabia.
Received: 03/02/2016 Revised: 16/02/2016 Accepted: 18/02/2016
ABSTRACT
In This paper develop the Analytic geometry of classical gauge theories, on compact dimensional manifolds, some important properties of fields k , the manifold structure M C of the configuration space, we study the problem of differentially projection mapping parameterization system by constructing C rank n on surfaces n 1 dimensional is sub manifold space Rn1 .
Index Terms: basic notion on differential geometry - differential between surfaces M , N R is called the differential manifolds- tangent and cotangent space- differentiable injective manifold- Operator geometric on Riemannian manifolds.
INTRODUCTION second countable is locally Euclidean space The object of this paper is to We introduce the concept of maximal C familiarize the reader with the basic analytic atlas, which makes a topological manifold of and some fundamental theorem in into a smooth manifold, a topological deferrable Geometry. To avoid referring to manifold is a Hausdorff, second countable is previous knowledge of differentiable local Euclidean of dimension n . If every manifolds, we include surfaces, which point p in M has a neighborhoodU such contains those concepts and result on that there is a homeomorphism from U differentiable manifolds which are used in onto a open subset of Rn . We call the pair a an essential way in the rest of the. The first coordinate map or coordinate system onU . section II present the basic concepts of We said chart (U,) is centered at p U , analytic Geometry (Riemannian metrics, ( p) 0 , and we define the smooth maps Riemannian connections, geodesics and curvature). consists of understanding the f : M N where M , N are differential relationship between geodesics and manifolds we will say that f is smooth if curvature, Jacobi fields an essential tool for there are atlases (U ,h ) on M and this understanding, are introduced in we (V , g ) on N . introduce the second fundamental from associated with an isometric immersion and NOTIONS ON DIFFERENTIAL prove a generalization of the theorem of GEOMETRY Riemannian Geometry this allows us to real 2.1 Basic analytic geometry the notion of curvature in Riemannian Definition 2.1.1 manifolds to the classical concept of A topological manifold M of Gaussian curvature for surfaces. way to dimension n , is a topological space with the construct manifolds, a topological manifolds following properties: C analytic manifolds, stating with topological manifolds, which are Hausdorff
International Journal of Research & Review (www.gkpublication.in) 78 Vol.3; Issue: 2; February 2016 (i) M is a Hausdorff space . For ever pair of Let M be a topological space n - points p, g M , there are disjoint open manifold. If (U,),(V, ) are two charts such subsets U,V M such that p U and g V . thatU V , the composite map 1 (ii) M is second countable . There exists (1) : (U V ) (U V ) accountable basis for the topology of M . Is called the transition map from to . (iii) M is locally Euclidean of dimension n . Definition 2.1.9 Every point of M has a neighborhood that A smooth structure on a topological ishomeomorphic to an open subset of R n . manifold M is maximal smooth atlas. Definition 2.1.2 (Smooth structures are also called A coordinate chart or just a chart on differentiable structure or C structure by a topological n manifold M is a pair (U,) , some authors). Where U is an open subset of M and ~ Definition 2.1.10 :U U is a homeomorphism from U to an A smooth manifold is a pair (M , A), ~ n open subset U (U) R . where M is a topological manifold and A is Examples 2.1.3 smooth structure on M . When the smooth Let S n denote the (unit) n sphere, structure is understood, we omit mention of which is the set of unit vectors in Rn1 : it and just say M is a smooth manifold. S n {x Rn1 : x 1} with the subspace Definition 2.1.11 topology, S n is a topological n manifold. Let M be a topological manifold. Definition 2.1.4 (i) Every smooth atlases for M is contained The n dimensional real (complex) in a unique maximal smooth atlas. (ii) Two smooth atlases for M determine the projective space, denoted by Pn (R) or Pn (C)) , is defined as the set of 1-dimensional linear same maximal smooth atlas if and only if their union is smooth atlas. subspace of R n1 or C n1) , P (R) or P (C) is a n n Definition 2.1.12 topological manifold. Let M be a smooth manifold and let Definition 2.1.5 p be a point of M . A linear map For any positive integer n , the n X :C (M) R is called a derivation at p if it torus is the product space T n (S 1 ... S 1 ) .It satisfies: is an n dimensional topological manifold. (2) X ( fg) f ( p)Xg g( p)Xf (The 2-torus is usually called simply the torus). Forall f , g C (M ) . The set of all derivation Definition2.1.6 of C (M) at p is vector space called the The boundary of a line segment is tangent space to M at p , and is denoted by [ the two end points; the boundary of a disc is Tp M ]. An element of Tp M is called a tangent a circle. In general the boundary of an n vector at p . manifold is a manifold of dimension (n 1) , Lemma 2.1.13 we denote the boundary of a manifold M as Let M be a smooth manifold, and M . The boundary of boundary is always suppose p M and X Tp M If f is a cons and empty, M . function, then Xf 0 . If f ( p) g( p) 0 , then Lemma 2.1.7 X ( fp) 0 . (i) Every topological manifold has a Definition2.1.14 countable basis of Compact coordinate If is a smooth curve (a continuous balls. map : J M , where J R is an interval) in a (ii) Every topological manifold is locally compact. smooth manifold M , we define the tangent vector to at t J to be the vector Definitions 2.1.8 d d (t ) |t T (t ) M , where |t is the dt dt
International Journal of Research & Review (www.gkpublication.in) 79 Vol.3; Issue: 2; February 2016 standard coordinate basis for T R . Other call them Convectors. If V * then :V R t common notations for the tangent vector to for the any vV , we denote the value of on v by v or by v, . Addition and d d are (t ), (t ) and | . This tangent tt dt dt multiplication by scalar in V * are defined by vector acts on functions by: the equations: d d d( f ) v v v , v v (3) (t ) f | f | f (t ) . 1 2 1 2 t t dt dt dt Where v V ,, V and R . Definition 2.1.15 Proposition2.3.1 Let V and W be smooth vector fields Let V be a finite- dimensional vector space. on a smooth manifold M . Given a smooth If (E1,...,En ) is any basis for V , then the function f : M R , we can apply V to f and 1 n convectors ( ,..., ) defined by. obtain another smooth function Vf , and we 1 if i j (5) i i , (E j ) j can apply W to this function, and obtain yet 0 if i j another smooth function W V f W(Vf ) . The Form a basis forV , called the dual basis to operation f W V f , however, does not in (E j ) .Therefore, dimV dimV . general satisfy the product rule and thus Definition 2.3.2 cannot be a vector field, as the following for ACr Convector field on M , r 0 , example shows let and on M V W is a function which assigns to each a x y convector p TP M in such a manner that R n , and let f (x, y) x, g(x, y) y . Then direct for any coordinate neighborhood U , with computation shows that V W( f g) 1, while
coordinate frames E1,..,En , the functions f V W g g V W f 0 , soV W is not a E , i 1,.....,n, are of class C r on U . For derivation of C (R2 ) . We can also apply the i convenience, "Convector field” will mean same two vector fields in the opposite order, C convector field. obtaining a (usually different) function W V f 2.4 The Exponential Map Normal . Applying both of this operators to f and Coordinates subtraction, we obtain an operator We have already seen that there are [V,W]:C (M) C (M) , called the Lie bracket many differences between the classical of V and W , defined by Euclidean geometry and the general [V,W] f V W f WV f . This operation is Riemannian geometry in the large. In a vector field. The Smooth of vector Field is particular we have seen examples in which Lie bracket of any pair of smooth vector one of basic axioms of Euclidean geometry fields is a smooth vector field. no longer holds. Two distinct geodesic (real Lemma 2.1.16 lines) may intersect in more than one point. The Lie bracket satisfies the following The global topology of the manifold is identities for allV ,W , X (M ) . Linearity: responsible for this “failure”. In this we will [ aV bW , X ] a [V , X ] b [W , X ] define using the metric some special a,b R , collections to being Euclidean. Let M , g [ X ,aV bW ] a [X ,V ] b [X ,W ]. be Riemannian manifold and U , an open (i) Ant symmetry[V,W] [W,V] . coordinate neighborhood with coordinate (ii) Jacobi identity x1,...,xn .We will try to find a local change [V,[ W, X ]][ W,[ X,V] ][ X,[V,W] ] 0 in coordinate xi yi in which the [ f V,g W ] f g [ V,W ] ( f V g) W ( g W f ) V For f , g C (M) expression of the metric is as close are to 2.3 Convector Fields i j the Euclidean metric g0 i, j dy dy . Let Let V be a finite – dimensional vector qu , be the point with coordinate 0,...,0 space over R and let V * denote its dual via a linear we may as well assume that space. Then V * is the space whose elements gi j (q) i, j . We would like “spread” the are linear functions from V to R, we shall
International Journal of Research & Review (www.gkpublication.in) 80 Vol.3; Issue: 2; February 2016 above equality to an entire neighborhood of several vector variables vI ,...., vr ofV , linear q . To achieve this we try to find local in each separately.(i.e. multiline). The coordinates y J near q such that in these number of variables is called the order of new coordinates the metric is Euclidean up the tensor. A tensor field of order r on a to order one i,e . manifold M is an assignment to each point
gi j P M of a tensor P on the vector space gi, j (q) (q) yk (6) TP M , which satisfies a suitable regularity i,J i, j (q) 0 , :i, j,k g 0 r k k condition C ,C ,orC as P varies on M . y y We now describe a geometric way of Theorem 2.4.3 producing such coordinates using the With the natural definitions of geodesic flow .Denote as usual the geodesic addition and multiplication by elements of r R the set (V ) s of all tensors of order (r, s) on from q with initial direction X Tq (M ) . By V forms a vector space of dimension nrs . X (t) Not the following simple fact L q Theorem 2.4.4 X V . Hence, there exists a small The maps A and S are defined on neighborhood V ofTq (M ) , Such that, for r r M a C manifold and M the C any X V , the geodesic X q (t) is defined for covariant tensor fields of order r , and they all t 1 .we define the exponential map at satisfy properties there. In these cases of (c), F * : r N r M is the linear map induced by q . a C mapping F : M N . expq :V Tq (M ) M , X X q (1) Definition 2.4.5 The tangent space T (M ) is a q Let V be a vector space and V Euclidean space, and we can define are tensors. The product of and , denoted D (r) T (M ) , the open “disk” of radius r q q is a tensor of order r s defined by : centered at the origin we have the following (v1,...vr ,...vr1,....vrs ) (v1,....,vr ) (vr1,....,vrs ) . result centered at the origin .we have the The right hand side is the product of following result the values of and .The product defines a i i i In particular, dx X h , that is, dx r rs a mapping , of x V V . measures the change in theis coordinate of a Theorem 2.4.6 point as it moves from the initial to the The product r V o r V rs V just terminal point of X . The preceding formula a defined is bilinear and associative. If may thus be written. 1,...., n is a basis of. f f (7) df X dx1(X ) ... dxn X . Definition 2.4.7 a x1 a xn a a a Carton’s wedge product, also known This gives us a very good definition as the exterior Product, as the ant symmetric of the differential a function on U R n ; is a tensor product of cotangent space basis field of linear functions which at any point a elements dx dy 1/ 2 (dx dy dy dx) dy dx . of the domain of f assigns to each vector X a Note that, by definition, dx dx 0 . The a number. Interpreting X a as the differential line elements dx and dy are displacement of the n independent variables called differential 1-forms or 1-form; thus from a , that is, a as initial point and a h as the wedge product is a rule for construction terminal point. d f X a a approximates g 2-forms out of pairs of 1-forms. (linearly) the change in f between these Remark 2.4.8 p points. Let p be an element of p , p an Definition 2.4.2 element of q . Then (1) pq . A convector tensor on a vector space p q q p Hence odd forms ant commute and the V is simply a real valued vI ,....,vr of
International Journal of Research & Review (www.gkpublication.in) 81 Vol.3; Issue: 2; February 2016 wedge product of identical 1-forms will Definition 2.4.15 always vanish. A smooth structure on a Hausdorff Definition 2.4.9 topological space is an equivalence class of A topological space M is called atlases, with two atlases A and B being
(Hausdorff) if for all x, y M there exist equivalent if for Ui ,i A and V j ,j B xU y V open sets such that and and with U i V j then the transition map U V i Ui V j j Ui V j is a Definition 2.4.10 diffeomorphism (as a map between open A topological space M is second sets of Rn ). countable if there exists a countable basis Definition 2.4.16 for the topology on M . A smooth manifold M of dimension Definition 2.4.11 n is a topological manifold of dimension n A topological space M is locally together with a smooth structure. Euclidean of dimension n if for every point Definition 2.4.18 xM there exists on open set U M and A map F :M N is called a open set w R n so that U and W is (homeomorphism). diffeomorphism if it is smooth objective and 1 Definition 2.4.12 inverse F : N M is also smooth. A topological manifold of dimension Definition 2.4.19 n is a topological space that is Hausdorff, A map F is called an embedding if F is an second countable and locally Euclidean of immersion and homeomorphism onto its dimension N. image Definition 2.4.13 Definition 2.4.20 A smooth atlas A of a topological If F :M N is an embedding then space M is given by: F(M ) is an immersed sub manifoldsof N . Example 2.1.21 (i) An open covering U iI where Ui M The vector function as vector fields Open and M U iI i on R E , the function fi t is vector fields (ii) A family i :Ui Wi iI of r tua , t R is parameterizations u ui homeomorphism i onto opens subsets n on line a ai is vector as point on line L . Wi R so that if Ui U j then the map r r(t) (tu1 a1,tu2 a2,tu3 a3) U U U U is i i j j i j (Adiffoemorphism) Example 2.4.14 The stereographic is map on :S 2 N onto R2 the noth pole (0,0,1) p S 2 N, ( p) is defined to be the point at which line N and p intersects the xy-plan :S 2 N R is
“diffeomorphism” to do so write explicitly Fig.(2) : vector fields 1 in coordinates and solute for 2.5: Tangent space and vector fields Let C (M, N) be smooth maps from M and N and let C (M) smooth functions
on M is given a point p M denote,
C ( p) is functions defined on some open
neighborhood of p and smooth at p . Definition 2.5.1
Fig. (1): the diffeomorphism International Journal of Research & Review (www.gkpublication.in) 82 Vol.3; Issue: 2; February 2016 * (i) The tangent vector X to the curve f smooth Element of Tp M are called c : , M at t 0 is the map cotangent vectors or tangent convectors at c(0):C (c(0)) R given by the formula. p . d( f c) (i) For f :M R smooth the composition (8) X ( f )c(0)( f ) f Cc(0) dt t0 * df p Tp M Tf ( p)R R is called and A tangent vector X at p M is the (ii) referred to the differential of f .Not that tangent vector at t 0 of some curve * df p Tp M so it is a cotangent vector at p . :, M with (0) p this is i (ii) For a chart U,, x of M and p U X (0):C ( p) R . i n * dxi then dx i1 is a basis of Tp M in fact Remark 2.5.2 n p d A tangent vector at is known as a is the dual basis of i . dx liner function defined on C ( p) which i1 Definition 2.6.1 satisfies the (Leibniz property) The elements in the tensor product X ( f g ) X ( f )g f X (g) (9) r * * Vs V ....V V ...V are called f , g C ( p) tensors or r-contra variant, s- contra Differential 2.5.3 varianttensor. Given F C (M,N) and p M and Remark 2.6.2 X T M choose a curve :(,) M with p The Tensor product is bilinear and (0) p and (0) X this is associative however it is in general not possible due to the theorem about existence commutative that is T1 T2 T2 T1 in of solutions of liner first order ODEs , then general. consider the map Definition 2.6.3 F :T M T N r * p p F ( p) mapping T Vs is called reducible if it can be / X F* p (X ) (F ) (0) , this is liner map written in the form T V ...V L1 ... Ls between two vector spaces and it is 1 r for. V V ,Lj V * 1 i r ,1 j s independent of the choice of . i r for . Definition 2.5.5 Definition 2.6.4 i, j The liner map F* p defined above is Choose two indices where 1 i r ,1 j s called the derivative or differential of F at for any reducible tensor 1 2 r r 1 T V1 ...Vr L .... L let Ci T Vs1 p while the image F* p (X ) is called the push We extend this linearly to get a linear map forward X at p M Definition 2.5.6 j r r1 Ci :Vs Vs1 which is called tensor- Given a smooth manifold M a vector field contraction. V is a map V :M TM mapping Definition 2.6.5 p V ( p) V p and V is called smooth if it Let F :M N be a smooth map is smooth as a map from M to TM . between two smooth manifolds and X (M ) Isan R vector space for 0 w Tk N be a k covariant tensor field we Y, Z X (M ) , p M and * define a k covariant tensor field F w over a,b R ,( aY bZ) aV bZ p p p and for M by. f C (M ) ,Y X (M ) define f Y :M TM F *w v ,...,v w F v ,...,F v (11) p 1 k F p * p 1 * p k mapping. v1,...,vk TpM p ( f Y ) f ( p)Y * (10) p p In this case F w is called the pullback of 2.6: Cotangent smooth n-manifolds w by F . Let M be a smooth n-manifolds and Example 2.6.7 p M .We define cotangent space at p The tangent bundle section is T *M n m denoted by p to be the dual space of the function f :R R is differential or * tangent space at p :Tp (M ) f :TpM R ,
International Journal of Research & Review (www.gkpublication.in) 83 Vol.3; Issue: 2; February 2016 tangent map as point p on tangent felids (ii) Induced metric df ( p) is image df p . Let M , g be a Riemannian manifold and df (v) v ,C f (C), pC f (p)C f : N M , g an immersion where N is a p f ( p) smooth manifold ( that is f is a smooth map and f is injective ) then induced metric on N is defined .
f g p v,w g f ( p) f* (v) , f* (w) (14) : v,wTp N , p N
The induced metric S n sometimes
denoted g n from the Euclidean space Eucl S n1 2 n1 R and g Eucl by the inclusion i :S R Fig (4): differential tangent map is called the standard (or round) metric on 2.7: Integration of differential forms clearly i is an immersion .Consider 2 3 M w is well defined only if M is stereographic projection S R and 2 2 orient able dim(M ) n and has a partition denote the inverse map u : R S then * of unity and w has compact support and is a u g Eucl . Given the Riemannian metric for 2 differential n-form on M . R . M , g M , g Example 2.7.1 (iii) Product metricIf 1 1 , 2 2 are The circular helix on curve is parameters on two Riemannian manifolds then the product
M1 M 2 admits a Riemannian metric
g g1 g2 is called the product metric defined by . g(u u ,v v ) g (u ,v ) g (u ,v ) (15) 1 2 1 2 1 1 1 2 2 2 .
Where ui ,vi Tp i M i for i 1,2,....we use the
fact that Tp , p ( M 1 M 2 ) Tp M 1 Tp M 2 . 1 2 1 1 (iv)Warped product Suppose M , g , Definition 2.7.2 1 1 M , g A pair M , g of a manifold M equipped 2 2 are two Riemannian manifolds M M , g f 2 g with a Riemannian metric g is called a then 1 2 1 2 is the warped g , g Riemannian manifold. product of 1 2 or denoted M1, g1 f
Definition 2.7.3 M 2 , g2 where f :M1 R a smooth Suppose M , g is a Riemannian manifold positive function is. p M 2 and we define the length ( or norm ) g1 f g2 p p u2 u2 ,v1 v2 (16) 1 2 of a tangent vector vT M to be p g1 u1,v1 f p1 g2 v2 ,w2 p1 p1 v v,v g , , p Recall and the Definition 2.7.6 A smooth map f :M, g N,h angle v,w between v, wTp M v 0 w by between two Riemannian manifolds is v,w (12) cos(v,w) p . called a conformal map with conformal v w * 2 factor :M R if f h g .A Examples 2.7.4 conformal map preserves angles that is (i).Euclidean metric (canonical metric) v,w f* (v), f* (w) for all u,vTp M and n g Eucl on R . p M . g dxi dx j dx1 dx1 ... dxn dxn (13) Eucl i j Example 2.7.7 1 1 n n dx dx ... dx dx
International Journal of Research & Review (www.gkpublication.in) 84 Vol.3; Issue: 2; February 2016 S 2 R3 We consider stereographic (ii) For any p M there exists a 2 2 projection S / pn R . As stereographic neighborhood U of P which is projection is a diffeomorphism its inverse homeomorphism to an open subset V Rm . u : R S / pn is a conformal map. It follows (iii) M has a countable basis of open sets, from an exercise sheet that u is a conformal coordinate charts (U,) map with conformal factor (iv) is equivalent to saying that p M has a 2 2 (x, y) 2 /1 x y . open neighborhood U P homeomorphism Definition 2.7.8 to open disc Dm in Rm , axiom (v) says that A Riemannian manifold M , g is M can covered by countable many of such locally flat if for every point p M there neighborhoods , the coordinate chart (U,) exist a conformal diffeomorphism where U are coordinate neighborhoods or f :U V between an open neighborhood charts and are coordinate . U of p and V R n of f (p) . A homeomorphisms , transitions Definition 2.7.9 between different choices of coordinates are
Given two Riemannian manifold called transitions maps i j j i , which M , g and N,h they are called isometric are again homeomorphisms by definition , 1 n of there is a diffeomorphism f : M N we usually write p (x), :U V R such that f *h g such that a differ- as coordinates forU and morphism f is called an isometric. p 1(x), 1 :V U M as coordinates Remark 2.7.10 for U , the coordinate charts (U,) are In particular an isometrics coordinate neighborhoods, or charts , and f :(M, g) (M, g) is called an isometric of are coordinate homeomorphisms, transitions (M , g) . All isometrics on a Riemannian between different choices of coordinates are manifold from a group. called transitions maps i j j i which Definition 2.7.11 are again homeomorphisms by definition , (M, g),(N,h) Are called locally we usually x ( p), :U V Rn as a isometric if for every point p M there is parameterization U . A collection f :U V an isometric from an open A (i ,UiiI of coordinate chart with neighborhoodU of p in M and an open M iUi is called atlas for M . The f ( p) neighborhoodV of in N . transition maps i j a topological space M is Definition 2.7.12 called (hausdorff ) if for any pair p, q M , Suppose f :(M, g) (N,h) is an there exist open neighborhoods p U and * immersion then f is isometric if f h g . qU such thatU U for a topological Definition 2.7.13 space M with topology U can be written A bundle metric h on the vector as union of sets in , a basis is called a bundle ( E,M, ) is an element of countable basis is a countable set . * * E E which is symmetric and positive Definition 2.8.1 definite. Let X be a set a topology U for X is 2.8: Differentiable injective manifold collection of X satisfying. The basically an m-dimensional (i) And X are in U topological manifold is a topological space (ii) The intersection of two members of U is M which is locally homeomorphism to Rm in U definition is a topological space M is called . an m-dimensional (topological manifold) if (iii) The union of any number of members the following conditions hold. U is inU . The set X with U is called a (i) M is a hausdorff space. topological space the members U u are called the open sets. let X be a topological
International Journal of Research & Review (www.gkpublication.in) 85 Vol.3; Issue: 2; February 2016 space a subset N X with xN is called a charts with is M U i called an atlas for neighborhood of x if there is an open set M of topological manifolds. A topological U with x U N , for example if X a manifold M for which the transition maps metric space then the closed ball D (x) and i, j ( j i ) for all pairs i , j in the atlas the open ball D (x) are neighborhoods of are homeomorphisms is called a x a subset C is said to closed if X \ C is differentiable, or smooth manifold, the open transition maps are mapping between open Definition 2.8.2 subset of Rm , homeomorphisms between m A function f : X Y between two open subsets of R are C maps whose C U topological spaces is said to be continuous if inverses are also maps , for two charts i for every open set U of Y the pre-image and U j the transitions mapping. f 1(U) is open in X . 1 i, j ( j i ):i (Ui U j ) j (Ui U j ) Definition 2.8.3 And as such are homeomorphisms between Let X and Y be topological spaces these open of Rm , for example the we say that X and Y are homeomorphism 1 differentiability ( ) is achieved the if there exist continuous function such that mapping ( (~)1 ) and (~ 1 ) which are f g id and g f id we write X Y y X homeomorphisms since (A A) by and say that f and g are homeomorphisms assumption this establishes the equivalence between X and Y , by the definition a (A A) , for example let N and M be function f :X Y is a homeomorphisms if smooth manifolds n and m and only if . respecpectively, and let f :N M be (i) f is a objective. smooth mapping in local coordinates (ii) f is continuous f f 1 :(U ) (V ) ,with respects (iii) f 1 is also continuous. charts (U,) and (V, ) , the rank of f at Definition 2.8.4 A differentiable manifold of p N is defined as the rank of f at ( p) dimension N is a set M and a family of i.e. rk ( f ) p rk (J f ) ( p) is the n injective mapping x R M of open sets Definition 2.8.5 n n u R into M such that. Let I n be the identity map on R , (i) u x (u ) M then Rn , I is an atlas for Rn indeed, if n (ii)For any , with x (u ) x (u ) Rn U is any nonempty open subset of , then U, I is an atlas for U so every open (iii)the family (u , x ) is maximal relative to n Rn C conditions the pair (u , x ) or the mapping subset of is naturally a manifold. Example 2.8.6 x with p x (u ) is called a parameterization , or system of coordinates The n-space is a manifold of dimension n when equipped with the atlas of M , u x (u ) M the coordinate charts A1 (Ui ,i ),(Vi , i ), 1 i n 1 where for (U,) where U are coordinate each 1i n 1. neighborhoods or charts, and are n coordinate homeomorphisms transitions are Ui ( x1,....,xn1)S , x1 0 i (x1,.....,xn1) between different choices of coordinates are (17) ( x1,..., xi1, xi1,...,xn1) called transitions maps.
: 1 i, j j i OERATOR GEOMETRIC ON Which are anise homeomorphisms RIEMANNIAN MANIFOLDS by definition, we usually write 3.1 Vector Analysis one Method Lengths] x ( p), :U V R n collectionU and Classical vector analysis describes p 1 (x), 1 :V U M for coordinate one method of measuring lengths of smooth
International Journal of Research & Review (www.gkpublication.in) 86 Vol.3; Issue: 2; February 2016 paths in R3 if v :0,1 R3 is such a paths, back a metric on S, g S i g g / S . For then its length is given by length v v(t)dt . example, any invertible symmetric n n Where v is the Euclidean length of the matrix defines a quadratic hyper surface in n n tangent vector (t) , we want to do the same R by H A x R ,(Ax , x ) 1where , n thing on an abstract manifold, and we are denotes the Euclidean inner on R , H A has clearly faced with one problem, how do we a natural. make sense of the length v (t) obviously , this Example 3.1.4 problem can be solved if we assume that The Poincare model of the hyperbolic plane there is a procedure of measuring lengths of is the Riemannian manifold D, g where tangent vectors at any point on our manifold D is the unit open disk in the plan and The simplest way to do achieve this is to the metric g is given by. 1 assume that each tangent space is endowed (18) 2 2 g 2 2 dx dy with an inner product. (Which can vary 1 x y point in a smooth). Example 3.1.6 Consider a lie group G , and denote by L Definition 3.1.1 G A Riemannian manifold is a pair (M .g ) its lie algebra then any inner product , consisting of a smooth manifold M and a on LG induces a Riemannian metric metric g on the tangent bundle, i.e a smooth h , on G defined by. g symmetric positive definite tensor field on h (x, y) x, y L1 X ,(L1)Y g g g g M . The tensor g is called a Riemannian (19) :g G, X y Tg (G) metric on M . Two Riemannian manifold 1 Where (L ) :T (G) T (G) is the are said to be isometric if there exists a g g 1 differential at g G of the left translation diffeomorphism :M1 M 2 such that 1 map Lg . One checks easily that check easily : g1 g2 If (M .g ) is a Riemannian that the correspondence G g , is a manifold then, for any xM the restriction smooth tensor field, and it is left invariant gx :Tx (M1)Tx (M2 ) R . Is an inner (i,e) Lg h h g G . If G is also product on the tangent space Tx (M ) we will frequently use the alternative notation compact, we can use the averaging (,) g (,) the length of a tangent vector technician to produce metrics which are x x both left and right invariant. vT (M) is defined as usual x 3.2 The Levi-Cavite Connection v g v,v1/ 2 . If v:a,b M is a piecewise x x To continue our study of smooth path, then we defined is length by Riemannian manifolds we will try to follow b L(v) v(t ) dt . If we choose local a close parallel with classical Euclidean a geometry the first question one may ask is 1 n coordinates ( x ,...., x ) on M , then we get a whether there is a notion of “straight line” local description of g as. on a Riemannian manifold. In the Euclidean 3 i j space R there are at least ways to define a (18) g g dx ,dx ,g g , i j i j x x i j line segment a line segment is the shortest Proposition 3.1.2 path connecting two given points a line Let be a smooth manifold, and segment is a smooth path v :0,1 R3 denote by RM the set of Riemannian metrics satisfying v(t) 0 . Since we have not said on M then RM is a non –empty convex cone anything about calculus of variations which in the linear of symmetric tensor deals precisely with problems of type. Example 3.1.3 (i) We will use the second interpretation as Let M , g be Riemann manifold and our starting point, we will soon see however S M a sub manifold if S M , denotes that both points of view yield the same the natural inclusion then we obtain by pull conclusion.
International Journal of Research & Review (www.gkpublication.in) 87 Vol.3; Issue: 2; February 2016 (ii) Let us first reformulate as know the called semi simple if its lie algebra is semi tangent bundle of R3 is equipped with a simple. natural trivialization, and as such it has a Proposition 3.2.4 natural trivial connection 0 defined by. Let M , g and q M as above .Then there 0 i j 0 :i, j Where, exists r 0 such that the exponential map.( 0 exp q:Dq (r) M Is a diffeomorphism on (19) j 0,i, j j ,i x i i to. The supermom of all radii r with this All the Christ off symbols vanish, moreover, property is denoted PM (q) . if g0 denotes the Euclidean metric, then. 0 0 0 0 Definition 3.2.5 i g , g , g , 0 (20) 0 j k i j k 0 i j k 0 j k 0 The positive real number PM (q) is called V (t)V (t) 0 the infectivity radius of M at q the infemur. So that the problem of defining P inf P (q) “lines” in a Riemannian manifold reduces to M q M choosing a “natural” connection on the Is called the infectivity radius of M tangent bundle of course, we would like this connection to be compatible with the metric Lemma 3.2.6 0T (M ) but even so, there infinitely many The Freshet differential at q of the connections to choose from. The following exponential map, D0 exp q :Tq (M ) T exp q (0)M Tq (M ) . fundamental result will solve this dilemma. Is the identity Tq (M ) Tq (M ) Proposition 3.2.1 Theorem 3.2.7 Let M, g be a Riemannian manifold for Let q,r and as in the previous and any compact subset TM there exists consider the unique geodesic r :0,1 M of 0 such that for any x, X k there length , joining two points Br (q) .if exists a unique geodesic V VX X :, M w:0,1 M is a piecewise smooth path such thatV(0) x,V(0) X with the same endpoint as then. One can think of a geodesic as 1 1 (23) (t) dt w(t) dt defining a path in the tangent bundle 0 0 t V (t),V (t) . The above proposition With equality if and only if shows that the geodesics define a local flow w(0,1)(0,1Thus ɤ is the shortest path, onT(M ) by joining its endpoints. t 3.4: Riemannian Geometry (21) x.X V (t),V (t) ,VX X Definition 3.2.2 Definition 3.4.1 Riemannian Metrics The local flow defined above is called the Differential forms and the exterior geodesic flow the Riemannian manifold derivative provide one piece of analysis on manifolds which, as we have seen, links in M, g when the geodesic low is global with global topological questions. There is flow i,e anyV X is defined at each moment X much more on can do when on introduces a of t for any x, X T(M) , then the Riemannian metric. Since the whole subject Riemannian manifold is call geodetically of Riemannian geometry is a huge to the use complete . of differential forms. The study of harmonic Definition 3.2.3 from and of geodesics in particular, we Let L be finite dimensional real lie algebra, ignore completely hare questions related to the killing paring or form is the bilinear curvature. map. Definition 3.4.2 Metric Tensor K:L L R,KX,Y tr (ad(X ).ad (Y)) In informal terms a Riemannian (22) :X,Y L metric on a manifold M is a smooth The lie algebra is said to be semi simple varying positive definite inner product on T if killing paring is a duality, a lie group G is tangent space x . To make global sense of this note that an inner product is a bilinear
International Journal of Research & Review (www.gkpublication.in) 88 Vol.3; Issue: 2; February 2016 * form so at each point x , we want a vector Dp (X )T (M ) so that (X ) a ( Dp (x)) * * in tensor product. Tx Tx We can put, just as defines a conical a conical 1-form on we did for exterior forms a vector bundle * T (M ) in coordinates ( x, y ) yi dy the * * * * i striation on. T M T M T Tx . The xM x projection p is p(x, y) x so if conditions we need to satisfy for a vector X a b so if given take i i bundle are provided two facts we used for xi yi the bundle of p-forms each coordinate the exterior derivative w d dxi dyi x ,...... , x system 1 n defiance a basis which is the canonical 2-from on the * dx1,...... ,dxn for each Tx in the coordinate cotangent bundle it is non-degenerate, so neighborhood and the n2 element. that the map X (i w) from the tangent * dxi dx j 1 i, j n . Given a corresponding bundle of T (M) to its contingent bundle is
* * basis for Tx Tx . The Jacobean of a change isomorphism. Now suppose f is smooth of coordinates defines an invertible linear function and T * (M ) its derivative is a 1- * * transformation. J : Tx Tx And we have a form do. Because of the isomorphism a corresponding. above there is a unique vector field X on * * * * * T (M) such that df (i w) from the g (24) J J Tx Tx Tx Tx Definition 3.4.3 Local Coordinate another function with vector field Y , System A Riemannian metric on manifold Definition 3.4.7 * M * * The vector field X on T (M) given by is a section g of Tx Tx which at each I w dH point is symmetric and positive definite. In a i is called the geodesist flow of the local coordinate system we can write. metric g . (25) g gij x dxidx j i, j Definition 3.4.11 If :a,bT * M Is an integral curve of the Where gi, j x g j,i x and is a smooth geodesic flow. Then the curve P in (M ) function, with g x positive definite. Often i, j is called ageodesic. In locally coordinates, if the tensor product symbol is omitted and the geodesic flow. g g x dx dx one simply writes. ij i j i, j (27) X a b i x j y Definition 3.4.4 i j A diffehomorphism F : M N , between two Tp M at every point p of M , thenTpM is Riemannian manifolds is an isometric if isomorphic to M Rn m here isomorphic F * g g N N means that TM and M R n are Definition 3.4.5 homeomorphism as smooth manifolds and Let M a Riemannian manifold and for every p M , the homeomorphism :0,1 M a smooth map i,e a smooth restricts to between the tangent space T p M curve in M . The length of curve is n 1 and vector space Pi R . L( ) g ( , ) dt . Where d , (t)D t 0 dt with this definition, any Riemannian GET PEER REVIEWED manifold is metric space define. The basic notions on analytic (26) d(x, y) inf L( ) R: (t) y geometry knowledge of calculus, including are Riemannian an manifold space. the geometric formulation of the notion of Proposition 3.4.6 the differential and the inverse function Consider any manifold and its cotangent theorem. The differential Geometry of bundle T *(M ) , with projection to the base surfaces with the basic definition of p :T * (M ) M , let X be tangent vector to differentiable manifolds, starting with
* * properties of covering spaces and of the T (M ) at the point Ta M then
International Journal of Research & Review (www.gkpublication.in) 89 Vol.3; Issue: 2; February 2016 fundamental group and its relation to Higher eduationpress and international covering spaces. press Beijing-Boston, ALM10, pp. 241- 264. REFERENCES 8. S.Robert. Strichaiz, Analysis of the 1. Osman. MohamedM, Basic integration laplacian on the complete Riemannian on smooth manifolds and application manifolds. Journal of functional maps with stokes theorem, analysis52, 48, 79(1983). http//www.ijsrp.org-6-januarly2016. 9. P.Hariujulehto. P.Hasto, V.Latvala, 2. sman.Mohamed M, fundamental metric O.Toivanen, The strong minmum tensor fields on Riemannian geometry principle for quasisuperminimizers of with application to tangent and non-standard growth, preprint submitted cotangent, http//www.ijsrp.org- to Elsevier, june 16, 2011.-Gomez, F, 6januarly2016. Rniz del potal 2004. 3. Osman. Mohamed M, operate theory 10. Cristian, D.Paual, The Classial Riemannian differentiable manifolds, Maximum principles some of ITS http//www.ijsrp.org- 6januarly2016. Extensions and Applicationd, Series on 4. J.Glover, Z, pop-stojanovic, M.Rao, Math. And it Applications, Number 2- H.sikic, R.song and Z.vondracek, 2011. Harmonic functions of subordinate 11. H.Amann, Maximum principles and killed Brownian motion, Journal of principlal Eigenvalues, J.Ferrera. J. functional analysis 215(2004)399-426. Lopez 5. M.Dimitri, P.Patrizia, Maximum 12. J.Ansgarstaudingerweg 9.55099 mainz, principles for inhomogeneous Ellipti Germany e-mail: inequalities on complete Riemannian [email protected]. De, Manifolds, Univ. deglistudi di Perugia, U.Andreas institute fiir mathematic, Viavanvitelli 1,06129 perugia, Italy e- MA6-3, TU Berlin strabe des 17.juni. mails: Mugnai@unipg,it , 24July2008. 136, 10623 Berlin, Germany, e-mail: 6. Noel.J.Hicks. Differential Geometry, [email protected]. Van Nostrand Reinhold Company450 13. R.J. Duffin, The maximum principle west by Van N.Y10001. and Biharmoni functions, journal of 7. L. Jin, L. Zhiqin, Bounds of math. Analysis and applications 3.399- Eigenvalues on Riemannian Manifolds, 405(1961).
How to cite this article: Osman MM. Analytic differential geometry with manifolds. Int J Res Rev. 2016; 3(2):78-90.
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International Journal of Research & Review (www.gkpublication.in) 90 Vol.3; Issue: 2; February 2016