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Analytic Differential Geometry with Manifolds 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.
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