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  • Form  A mapping from a vector space
  • Connection form

    2. 2. A connection form is a manner of organizing data of a connection language of moving frame and differential form
    3. 3. Moving Frame <ul><li>Moving frame is a flexible generalization of the notion of a ordered basis of vector space often used to study the exterior differential geometry of smooth manifolds embedded in a homogeneous space. </li></ul>
    4. 4. Differential Form <ul><li>Differential form is a mathematical concept in fields of a multivariative calculus differential topology and tensors. </li></ul>
    5. 5. Moving Frame <ul><li>1.PROJECTIVE FRAME </li></ul><ul><li>2.AFFINE FRAME </li></ul><ul><li>3.LINEAR FRAME </li></ul><ul><li>4. ORTHONORMAL FRAME </li></ul><ul><li>5. EUCLIDEAN FRAME </li></ul><ul><li>6. CO-FRAME </li></ul><ul><li>7. KLEIN GEO-FRAME </li></ul><ul><li>8. FRENET-SERRET FRAME </li></ul><ul><li>9.DARBOAX FRAME </li></ul><ul><li>10.DESCRETE FRAME/K-THEORY </li></ul><ul><li>11.TIME </li></ul>
    6. 6. Projective Frame <ul><li>1. Sierpinki Space / Homology </li></ul><ul><li>2. Pre-regular Space/ Cross Ratio </li></ul><ul><li>3. Second Countable Space /Quadrangle Theorem </li></ul><ul><li>4. Normal Space /Harmonic Range </li></ul><ul><li>5. Lindelof Space/Paskals Theorem </li></ul><ul><li>6. Functional Space/ Pappus Theorem </li></ul><ul><li>7. Fully Normal Space / Brain Chon </li></ul><ul><li>8. Counter Space </li></ul>
    7. 7. Pre–regular Space (SUNDARANAND) <ul><li>X is a pre-regular space if any two topological distinguasible points can be separated by neighborhood. </li></ul>
    8. 8. Homology ( SUNDARANAND) <ul><li>A basic projective transformation is which corresponding slides meet on a fixed line called axis and corresponding points lie on a line through the centre. </li></ul>
    9. 9. Functional Space ( SARBANAND) <ul><li>Functional space is A set of A given kind from A set X to A set Y. </li></ul>
    10. 10. Cross Ratio ( SARBANANDA) <ul><li>Cross ratio of four points is only numerical invariant of projective geometry. </li></ul>
    11. 11. Second Countable Space ( CHANDRASEKHAR) <ul><li>Satisfy the second axiom of count ability. </li></ul><ul><li>Has a comfortable base. </li></ul><ul><li>Is separable and lindelof. </li></ul>
    12. 12. Quadrangle Theorem (Chandrashekhar) <ul><li>If two quadrangles have 5 pairs of corresponding sides meeting in collinear points the sixth pair meets on the same line. </li></ul>
    13. 13. Normed Space (AMBER) <ul><li>CONSIST OF T4,T5 &T6 </li></ul>
    14. 14. Harmonic Range ( AMBER) <ul><li>Construction of two pairs of points harmonically have cross ratio 1. </li></ul>
    15. 15. Lindelof Space (KAPILAMBER) <ul><li>Is a topological space in which every open cover has a countable sub cover. </li></ul><ul><li>More commonly used notion of compactness. </li></ul><ul><li>Strongly lindof / suslin. </li></ul>
    16. 16. Pascal's Theorem ( KAPILAMBAR)
    17. 17. Fully Normal Space ( VAB) <ul><li>Is a topological space in which every open cover admits an open locally finite refinement. </li></ul>
    18. 18. Pappus’s Theorem ( VAB)
    19. 19. Sierpinki Space (BAMAN) <ul><li>It is a smallest example of a topological space which is neither trivial nor discrete. </li></ul><ul><li>Is a finite topological space with two points, only one of which is closed. </li></ul>
    20. 20. Brain Chon (BAMAN)
    21. 21. Counter Space (BHIRUK)
    22. 22. Affine Frame <ul><li>An affine space is a set with a faithful freely transitive vector space action i,e a tensor for the vector space </li></ul><ul><li> :SXS  (a,b)|  (a,b) </li></ul>
    23. 23. Affine Frame <ul><li>1. QUOTENT SPACE / David Kay’s Axiom </li></ul><ul><li>2. FRECHET SPACE (T1) / David Kay’s Axiom ( AS1) </li></ul><ul><li>3. HOUSDORF SPACE(T2) / David Kay’s Axiom ( AS2) </li></ul><ul><li>4. REGULAR SPACE(T3) / David Kay’s Axiom ( AS3) </li></ul><ul><li>5. NORMAL SPACE(T4) / David Kay’s Axiom ( AS4) </li></ul><ul><li>6. NORMAL SPACE(T5) / David Kay’s Axiom ( AS5) </li></ul><ul><li>7. NORMAL SPACE(T6) / David Kay’s Axiom ( AS6) </li></ul>
    24. 24. Quotient Space (BHUTESH) <ul><li>Identification space is intuitively speaking the result of identifying or “gluing together” certain points of a given space. </li></ul><ul><li>Equivalence relation </li></ul>
    25. 25. David Kay’s Axim (BHUTESH) <ul><li>David Kay’s description of 3-dimensional affine space is as follows…. </li></ul><ul><li>“ An affine space is any system of points ,lines and planes which satisfy 6 axioms.” </li></ul>
    26. 26. T1 Space, Frechet Space (SANGBART) <ul><li>It is complete as a uniform space. </li></ul><ul><li>It is locally convex. </li></ul>
    27. 27. ASI (SANGBARTA) <ul><li>Two distinct points determine a unique line. </li></ul>
    28. 28. T2 Space, Hausdorff Space (BIKRITAKH) <ul><li>A topological space in which points can be separated by neighborhood. </li></ul>
    29. 29. AS2 (BIKRITAKH) <ul><li>Three non-linear points determine a unique plane. </li></ul>
    30. 30. T3 Space (SANGHAR) <ul><li>Normal space </li></ul>
    31. 31. AS3 ( SANGHAR) <ul><li>If two points lie in a plane then the line determined by these points lies in that plane. </li></ul>
    32. 32. T4 Space (DANDAPANI) <ul><li>Normal space </li></ul>
    33. 33. AS4 (DANDAPANI) <ul><li>If two planes meet their intersection is a line. </li></ul>
    34. 34. T5 space (Chakrapani) <ul><li>Normal space </li></ul>
    35. 35. AS5 ( CHAKRAPANI) <ul><li>There exist at least four non-linear points and at least one plane. </li></ul><ul><li>Each plain contains at least three non-linear points. </li></ul>
    36. 36. T6 Space (Kal) <ul><li>Normal space </li></ul>
    37. 37. AS6 ( KAL) <ul><li>Given any two non co-planer lines, there exist a unique plane through the first line which is parallel to second line. </li></ul>
    38. 38. Linear Frame <ul><li>1. CO-ODINATE SPACE. </li></ul><ul><li>2. SYMMETRIC SPACE/ Standard Basis. </li></ul><ul><li>3. CONFORMAL SPACE /Hamel Basis. </li></ul><ul><li>4. DUAL SPACE. </li></ul><ul><li>5. METRIC SPACE /Orthonormal Base. </li></ul><ul><li>6. LUSINS SPACE /Schander Base. </li></ul><ul><li>7. ALJEBIC DUAL SPACE. </li></ul><ul><li>8. ANTI DE-SITTER SPACE. </li></ul>
    39. 39. Co-ordinate Space ( Kamadiswar) <ul><li>Co- Ordinate space is a proptotypical example of n-dimensional vector space over a field F. </li></ul>
    40. 40. Symmetric Space (AMRITAKH) <ul><li>Same as R0 space </li></ul>
    41. 41. Standard Basis ( Amritakh) <ul><li>Standard basis is a sequence of or the unit vector. </li></ul><ul><li>Standard basis of a n-dimensional Euclidean space R n is the basis obtained by taking the n-basis vector. </li></ul><ul><li> e i : 1  I  n  </li></ul><ul><li>Where e i is the vector with a 1 in the co-ordinate and 0 elsewhere. </li></ul>
    42. 42. Algebraic Dual Space (KAPALI) <ul><li>Given any vector space V over some field F we define dual space V* to be the set of linear function V , i,e scaler valued linear map on V. </li></ul>
    43. 43. Conformal Space (BOMKESH) <ul><li>Conformal geometry is the study of the set of angle-preserving (conformal) transformation on a Riemann manifold. </li></ul>
    44. 44. Hamel Basis (BOMKESH) <ul><li>Where the number of tensors in the linear combinations </li></ul><ul><li>A 1 v 1 +……..+a n v n </li></ul><ul><li>Is always finite. </li></ul>
    45. 45. Metric Space (TRIPURESH) <ul><li>Metric space is a set where a notion of distance (metric) between elements of the set is defined. </li></ul>
    46. 46. Orthonormal Basis (TRIPURESH) <ul><li>Orthonormal basis of an inner product space V is a set of basis vectors whose elements are mutually orthonormal and of magnitude 1. </li></ul>
    47. 47. Dual Space (ESWAR) <ul><li>Defined a finite-dimensional vector space can be used for defined tensors which are studied in tensor space, </li></ul>
    48. 48. Dual Space (ESHAWAR) <ul><li>Given any vector V over some field F we define dual space V* to be the set of linear function V, i,e scaler valued linear map on V. </li></ul>
    49. 49. Lusin Space (NAKULISH) <ul><li>Lusin space is topological space such that some weaken topology makes it into a Polish space. </li></ul>
    50. 50. Schander Basis (NAKULESH) <ul><li>A scander basis is similar to Hamel basis . The difference is that for Hamel basis , linear combination are assured to be finite sums while for schauder basis they may be infinite. </li></ul>
    51. 51. Anti De-sitter Space (KHIROKANTHHA) <ul><li>Anti de-sitter space can be visualized as the lorentizian analogs of a sphere in a space of an additional dimension. </li></ul>
    52. 52. Orthonormal Frame <ul><li>1. FORCK SPACE / Multilinear Operation. </li></ul><ul><li>2. LP space / Bilinear Operation. </li></ul><ul><li>3. ORBIT SPACE / Sesquilinear Form. </li></ul>
    53. 53. Fock Space (UNMATT) <ul><li>Fork space is a algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particle. </li></ul><ul><li>Fork space is a Hilbert space made from direct sum of tensor product of single-particle Hilbert space. </li></ul>
    54. 54. Multilinear Operation (Unmatt) <ul><li>Multilinear operation is a map of type </li></ul><ul><li>f : V n  k </li></ul><ul><li>Where V is a vector space over field k, that is separate linear in each its N variable. </li></ul>
    55. 55. Lebesgue Space (LP)( LAMBAKARNA) <ul><li>P-form can be extended to vectors having an infinite number of components; yielding the space Lp. </li></ul>
    56. 56. Bilinear Operation ( Lambakarna) <ul><li>Bilinear is a function which is linear in both of its arguments. Let v , w and be three vector spaces over the same base field F. </li></ul><ul><li>A linear map B: VXW  X such that for any w in W the map v ם B (v, w) is a linear map from v to x and for any v in V the map w ם B( v , w ) is a linear map for w to x. </li></ul>
    57. 57. Orbit Space (BISHES) <ul><li>Suppose a topological group G acts continuously on a space X. One can form a equivalence relation on X by saying points are equivalent if and only if they lie in the same orbit . </li></ul><ul><li>The quotient space under this relation is called the Orbit space. </li></ul>
    58. 58. Sesquilinear Form ( BISHES) <ul><li>A Sesquilinear form on a complex vector space v is a map in one argument and anti-linear in other. </li></ul><ul><li>If  (x+y, z+w) </li></ul><ul><li>=  (x, y)+  (x, w)+  (y, z)+  (y, w) </li></ul><ul><li> (ax, by)= ãb  (x, y) </li></ul><ul><li>For all x, y, z,w in V AND a ,b in C </li></ul>
    59. 59. Euclidean Frame <ul><li>1. Baire Space </li></ul><ul><li>2.Topological Space. </li></ul><ul><li>3. Tychonoff Space. </li></ul>
    60. 60. T 3 ½ , Tychonoff Space (Trisangkeswar) <ul><li>Completely regular space. </li></ul>
    61. 61. Topological Space ( JAGANATH) <ul><li>Topological spaces are mathematical structures that allow the formal definition of concepts:- </li></ul><ul><ul><li>Convergence </li></ul></ul><ul><ul><li>Connectedness </li></ul></ul><ul><ul><li>Continuity </li></ul></ul>
    62. 62. Baire Space ( SAMBRANAND) <ul><li>Baire space is a very large and “enough” points for certain limit process of closed set with empty interior. </li></ul>
    63. 63. Co-frame <ul><li>1.SUSLIN SPACE/Dual Frame </li></ul><ul><li>2.SCHWART SPACE/Co-tangent Bundle </li></ul><ul><li>3.RANDOM SPACE/Kronker Delta </li></ul>
    64. 64. Suslin Space ( NIRMISH) <ul><li>Is a separable completely amortizable topological space. </li></ul><ul><li>I, e a space homogeneous to a complete metric space that has a comfortable dense subset. </li></ul><ul><li>A Suslin space is the image of a polish space under a continuous mapping. </li></ul>
    65. 65. Dual Frame (NIRMISH) <ul><li>A moving frame determines a dual frame of co-tangent bundle over U, which is sometimes also called Moving Frame. </li></ul><ul><li>This is a n-tuple of smooth1-form a 1 ,a 2 ,--a n which are linearly independent at each point q in U. </li></ul>
    66. 66. Radon Space (NANDIKESWAR) <ul><li>A Random space is a topological space such that every finite Boral measure is inner regular. </li></ul>
    67. 67. Co-tangent Bundle (NANDIKESHAR) <ul><li>Co-tangent bundle of a smooth manifold is the vector bundle of all the co-tangent space at every point in the manifold. </li></ul>
    68. 68. Schwartz Space (KRODHISH) <ul><li>Schwartz space is the function space of rapidly decreasing function. </li></ul>
    69. 69. Kroncker Delta ( KRODHISH) <ul><li>Kroncker delta is a function of two bundles which is 1 if they are equal, 0 if otherwise. </li></ul>
    70. 70. Darboux Frame <ul><li>1.CONTOUR SPACE/Adapted Frame </li></ul><ul><li>2.LOCALLY CONVEX SPACE/ Maurer – Cartan Frame </li></ul><ul><li>3.F SPACE /Push Forward & Pull Back </li></ul>
    71. 71. Contour Space (ABHIRUK) <ul><li>Each lip contour is a point in an 80-dimensional “ contour space”. </li></ul>
    72. 72. Push Backward & Forward (ABHIRUK) <ul><li>Let  :M  N be a smooth map of smooth manifolds. </li></ul><ul><li>Given some x  M, the Push forward of  at x is a linear map </li></ul><ul><li>d  x :T x M  T  (x) N from tangent space of M at x to the tangent space of N at  (x). </li></ul><ul><li>The applicant vector X is sometimes called the Push Forward of x by  . </li></ul>
    73. 73. Locally Convex Space ( BAKRANATH) <ul><li>Locally convex space is defined either in terms of convex set or equivalent in terms of semi norm. </li></ul>
    74. 74. Adapted Frame (BAKRANATH) <ul><li>Let  :M  E n be an embedding of p-dimensional smooth manifold into a euclidean space. </li></ul><ul><li>The space of adopted frame on M denoted by F  (M) is the collection of tuples (x 1 f 1 ,….f n ) where x  M and f 1 form an ortonormal basis of En such that f 1 ….f q are tangent to  (M) at  (v). </li></ul>
    75. 75. F-space (RAKHASESWAR) <ul><li>Is a vector space V over the real or complex number together with a metric </li></ul><ul><li>D:VXV  R </li></ul><ul><li>V is continuous transition invariant. </li></ul>
    76. 76. Maurer-cartan Form (RAKHASHESWAR) <ul><li>Let g = TeG be the tangent space of a Lie group G at identity G acts on itself by left translation </li></ul><ul><li>L:GXG  G such that g  G </li></ul><ul><li>We have Lg:G  G where Lg(h)=gh this induces a map of tangent bundle on itself (Lg)*Thg  TghG </li></ul><ul><li>Aleft invariant vector field is a section x of TG such that </li></ul><ul><li>(Lg)*X=X  g  G </li></ul><ul><li>The Mauer-Cartan form w is g-valued one-form on G defined on vectors v  TgG by formula </li></ul><ul><li>w(v)=(Lg -1 )*v </li></ul>
    77. 77. Klein Geometry <ul><li>Accessible Space </li></ul><ul><li>2. Sequential Space </li></ul><ul><li>3. De-sitter Space </li></ul><ul><li>4. Homogeneous Space </li></ul><ul><li>5. Symmetric Space </li></ul><ul><li>6. R0 Space </li></ul>
    78. 78. Sequential Space (VADRASEN) <ul><li>Sequential space /Frechet Urysohm space satisfy a very weak axiom of compatibility. </li></ul><ul><li>A sequential space is a space X satisfying one of the following equivalent conditions. </li></ul><ul><li>1. Every sequential open subset of X is open. </li></ul><ul><li>2. Every sequential closed subset of X is closed. </li></ul>
    79. 79. De-sitter Space (BHIMLOCHAN) <ul><li>De-Sitter space is Lorentz an analogue of an n-sphere (with its canonical Riemannian metric). </li></ul><ul><li>De-Sitter space can be defined as a sub manifold of Murkowski space in one higher </li></ul>
    80. 80. Homogeneous Space (BAKRAKUNDA) <ul><li>Traditional spaces are homogeneous space, but not for a uniquely determined group. </li></ul><ul><li>Changing the group changes the appropriate language. </li></ul>
    81. 81. Regular Space (BAIDYANATH) <ul><li>X is a T3 space if and only if it is both regular and Hausdorff. </li></ul>
    82. 82. R 0 Space (RURU) <ul><li>Let X be a topological space and let x and y be points in X . </li></ul><ul><li>We say that x and y can be separated if each lies in an open set which does not contain the other point X is a R 0 space if any two topologically distinguishable points in X can be separated. </li></ul>
    83. 83. Accessible Space ( ASITANGA) <ul><li>X is a T1 space if any two distinct points in X can be separated. </li></ul>
    84. 84. K-theory <ul><li>1.BANACH SPACE/ Representative K-theory </li></ul><ul><li>2.ADJUNCTION SPACE/Disjoint Union </li></ul>
    85. 85. Adjunction Space <ul><li>Suppose X is a space and A is a subspace of X . One can identify all points in A to a single equivalence class and leave points outside of A equivalent only to themselves. </li></ul><ul><li>The resulting quotient space is denoted X/A. </li></ul>
    87. 87. Banach Space (KAPALI) <ul><li>Banach space is a vector space V over the real or complex number with a nom !!.!!. </li></ul>
    89. 89. Franet-serret Frame <ul><li>1.Affine Space </li></ul><ul><li>2.Twistor Space </li></ul><ul><li>3.Cauchy Space </li></ul><ul><li>4.Euclidean Space </li></ul>
    90. 90. Affine Space (SHUR) <ul><li>An affine space is any system of points , lines and planes which satisfy 6 axioms. </li></ul>
    91. 91. Twister Space (DEBBAHU) <ul><li>Is a 4-dimensional complex space </li></ul><ul><li>T:=C 4 </li></ul><ul><li>It has associated to it the double fibration of flag manifold </li></ul><ul><li>P   F   M </li></ul><ul><li>Where projective Twister space </li></ul><ul><li>P:=F1(T)=P3(C)=P( C4) </li></ul>
    92. 92. Cauchy Space (SHIV) <ul><li>A Cauchy space is a set X and a collection of C of proper filters in power set P(X). </li></ul>
    93. 93. Euclidean Space (MAHADAR) <ul><li>An n-dimensional space with a notion of distance and angle that obey Euclidean relationship. </li></ul>
    94. 94. Differential Form <ul><li>Connection Form </li></ul><ul><li>2. Curvature Form </li></ul><ul><li>3. Torsion Form </li></ul><ul><li>4. Solder Form </li></ul><ul><li>5. Levi-Civita Connection Form </li></ul><ul><li>Lie Bracket Form </li></ul><ul><li>Descrete-1 </li></ul><ul><li>8. Descrete-2 </li></ul><ul><li>9. Descrete-3 </li></ul><ul><li>10. Descrete-4 </li></ul><ul><li>11. Base States </li></ul>
    95. 95. Connection Form (Shak) <ul><li>1. Principal Connection </li></ul><ul><li>2. Compatible Connection </li></ul><ul><li>3. Cartan Connection </li></ul><ul><li>4. Ehesmann Connection </li></ul><ul><li>5. Associated to Principal </li></ul><ul><li>6. Global Connection Form </li></ul><ul><li>7. Affine Connection </li></ul>
    96. 96. Principal Connection ( PURAJAB) <ul><li>Let  U  be an open cover of M, along with G-frame on each U, denoted by e U . </li></ul><ul><li>These one related on the intersection of overlapping open set by e V =e v .h uu for some G-valued function h uu defined in U n V. </li></ul>
    97. 97. Compatible Connection (PABMAN) <ul><li>A connection is compatible with the structure of a G-bundle on E provided that the associated parallel transport maps always sends one G-frame to another. </li></ul>
    98. 98. Cretan Connection (DHRUBHANIK) <ul><li>Cretan Connection are closely related to pseudo-group structure on a manifold. </li></ul><ul><li>Pseudo group is an extension of group concept , but one that grew out of the geometric approach of Sophus Lie. </li></ul>
    99. 99. Ehesmann Connection ( CHITRAREK) <ul><li>Ehesmann connection is a version of the notion of a connection which is defined on ordinary fiber bundle. </li></ul>
    100. 100. Associated To Principle ( BAHURUP) <ul><li>A Principle G-connection w in a principal G-bundle P  M gives rise to a collection of connection focus on M. </li></ul>
    101. 101. Global Connection Form ( BISHWAVAB) <ul><li>If  Up  is an open covering of M and each Up is equipped with a trivialization ep of E , then it is possible to define a global form in terms of the patches data between the local connection form on the overlap revision. </li></ul>
    102. 102. Affine Connection (MANAJAB) <ul><li>An affine connection is a geometrical object on a smooth manifold which connect near by tangent spaces and so permits tangent vector fields to be differentiable as if they were functions on the manifold with values in a fixed vector space. </li></ul>
    103. 103. Curvature Form (KUSH) <ul><li>1.Curvilinear Co-ordinate </li></ul><ul><li>2.Non-tensor </li></ul><ul><li>3.Parallel Transport </li></ul><ul><li>4.Geodesics </li></ul><ul><li>5.Co-variant Derivative </li></ul><ul><li>6.Curvature Tensor </li></ul><ul><li>7. Bianchi Identity </li></ul>
    104. 104. Curvilinear Co-ordinate (BASU) <ul><li>Curvilinear co-ordinates are co-ordinate in which angles between axes can changes from point to point. </li></ul>
    105. 105. Nontensors ( HARRUCHI) <ul><li>Nontensor is a tensor like quantity N  that behaves like a tensor in the raising and lowering of indices </li></ul><ul><li>N  = g  N  N  = g  N  </li></ul><ul><li>But does not transform like a tensor under a co-ordinate transformation. </li></ul>
    106. 106. Parallel Transport ( NAVIGUPTA)
    107. 107. Geodesics (SUTABRATA) <ul><li>Suppose we have a point x u that moves along a truck in physical space time. </li></ul><ul><li>Suppose the track is parameterized with the quantity  .The variation of velocity upon parallel displacement along the track can be calculated. </li></ul><ul><li>If there are no force acting on the point then the velocity is unchanged and we have Geodesics Equation. </li></ul>
    108. 108. Co-variant Derivative ( BIKIKAKT) <ul><li>The partial derivative of a vector w.r.t a space-time co-ordinate is composed of two parts. </li></ul><ul><li>Normal partial derivative minus the change in the vector due to parallel transport. </li></ul><ul><li>A  :  =A  1  _A a T  a </li></ul>
    109. 109. Curvature Tensor (BAMDEV) <ul><li>The curvature K of a surface is simply the angle through a vector is turned as we take it around an infinitesimal closed path. </li></ul>
    110. 110. Bianchi Identity (BASUDAN) <ul><li>Following differential relation known as Bianchi identity </li></ul><ul><li>R   :  +R   :  +R   :  =0 </li></ul>
    111. 111. Torsion Form ( KOUNCH) <ul><li>1. Curvature Tensor </li></ul><ul><li>2. Cyclic Sum </li></ul><ul><li>3. Curvature Form </li></ul><ul><li>4. Affinity Parameterized Geodesic </li></ul><ul><li>5. Twisting of Reference Frame </li></ul><ul><li>6. Torsion of a Filament </li></ul><ul><li>7. Affine Developments </li></ul>
    112. 112. Curvature Tensor (AM) <ul><li>Curvature tensor (  ) is a mapping </li></ul><ul><li>TM  TM  End (TM) </li></ul><ul><li>Defined on vector field x, y & z </li></ul><ul><li>R(X,Y)Z=  x  y Z-  y  x Z-   x, y  Z </li></ul>
    113. 113. Cyclic Sum ( MADHUBRAHA) <ul><li> (R(X,Y)X):= </li></ul><ul><li>R(X,Y)Z+R(Y,Z)X+R(Z,X)Y </li></ul>
    114. 114. Curvature Form (MEGHAPRISTA) <ul><li>Curvature form is the gl(n)-valued 2-form. </li></ul><ul><li> =Dw = dw +w  w </li></ul><ul><li>D denotes the exterior covariant derivative </li></ul><ul><li>1. D  =  </li></ul><ul><li>2. D  =0 </li></ul>
    115. 115. Affinely Parametrized Geodesic ( SUDHAMA) <ul><li>Suppose y(t) is a curve on M </li></ul><ul><li>Then  y (t) Y (t)=0 </li></ul><ul><li>For all time t is the domain of Y. </li></ul>
    116. 116. Twisting Of Reference Frame ( BHAJISTA) <ul><li>Frenet-Serret Formula describe how a particular moving frame twists along a curve. </li></ul>
    117. 117. Torsion Of A Filament ( LOHITAN) <ul><li>Length –maximizing (geodesic) configuration and its energy-minimizing configuration. </li></ul>
    118. 118. Affine Developments (BANASPATY) <ul><li>Suppose that x t is a curve in M.The affine developments of x t is the unique curve ct in Tx 0 M such that </li></ul><ul><li>Ċ t =  t 0 x t , c 0 =0 </li></ul><ul><li>Where  t 0 :Tx t M  Tx 0 M </li></ul><ul><li>is the parallel transport associated to  . </li></ul>
    119. 119. Solder Form (SHALMLI) <ul><li>1. Smooth Manifold </li></ul><ul><li>2. Symplectic Form </li></ul><ul><li>3. Complex Manifold </li></ul><ul><li>4. Vierbein or Tetrad Theory </li></ul><ul><li>5. Signature </li></ul><ul><li>6. Palatini Action </li></ul><ul><li>7. Interwiners </li></ul>
    120. 120. Smooth Manifold ( SURACHAN) <ul><li>A differentiable manifold for which all the transitions maps are smooth. </li></ul><ul><li>C  </li></ul>
    121. 121. Symplectic Form ( SOUMANAS) <ul><li>A symplectic form on a manifold M is a non-degenerative closed two form w. </li></ul>
    122. 122. Complex Manifold ( RAMANAK) <ul><li>Complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. </li></ul>
    123. 123. Vierbeib Or Tetrad Theory ( DEVBARS) <ul><li>Special case of application of Cretan connection in four – dimensional manifold. </li></ul>
    124. 124. Signature (PARIVADRA) <ul><li>Signature is the number of positive and negative Eigen values of the vector, </li></ul>
    125. 125. Palatini Action ( APAYAN) <ul><li>In tetrad formulation of general relativity, the action as a function of the co-tetrad e and a connection form A over a differential manifold M is given below where F is the gauge curvature 2-form and  is the anti symmetric inter winer of SO(3,1) normalized by  . </li></ul>
    126. 126. Interwiners (ABIGAN) <ul><li>If X and Y are the representative spaces of two linear representations of G has a linear map f:X  Y is called an interwiner of representations if it commutes with the action of G. </li></ul><ul><li>Thus an interwiner is an equivalent map in the special case of two linear representations/ action. </li></ul>
    127. 127. Levi-civita Connection (PLAKH) <ul><li>1. Preserve the metric </li></ul><ul><li>2. Derivative along curve </li></ul><ul><li>3. Unit sphere in R3 </li></ul><ul><li>4. Isomorphism </li></ul><ul><li>5. Leibniz Rule </li></ul><ul><li>6. Torsion Free </li></ul><ul><li>7. Weitzenbock Connection </li></ul>
    128. 128. Preserve The Metric ( SHIV) <ul><li>Preserve the metric I,e for any vector field x, y, z we have </li></ul><ul><li>X (g (Y,Z))=g (  x Y, Z) + g (Y,  x Z) </li></ul><ul><li>Where X (g (Y,Z)) denotes derivative of a function g (Y,Z) along the vector field X. </li></ul>
    129. 129. Derivative Along Curve (SUVADRA) <ul><li>Derivative along curve D = Pullback </li></ul><ul><li>Given a smooth curve Y on (M, g) and a vector field V along Y its derivative determined by </li></ul><ul><li>D t V =  ý (t) V </li></ul>
    130. 130. Unit Sphere In R 3 ( SHANT) <ul><li>Let S2 be the unit sphere in R3 .The tangent space to S2 at a point m is naturally defined with vector sub-space of R3 consisting of all vectors orthogonal to m. </li></ul>
    131. 131. Isomorphism (KHEM) <ul><li>Parallel transport along a curve w.r.t a connection define isomorphism between the tangent space at that point of the curve. </li></ul>
    132. 132. Leibniz RulE (AMRIT) <ul><li>C  (S 2 ) linear is first variable. </li></ul><ul><li>Consider a map f:S 2  R m  Y ( m ), m  </li></ul><ul><li>The map f is constant hence differential vanishes </li></ul><ul><li>dm f (x)=  dm Y (x) m  +  Y (m) ,X (m)  =0 </li></ul><ul><li> (  x Y) (m), m  =0 </li></ul>
    133. 133. Torsion Free ( ABHAY) <ul><li>Torsion free i,e for any vector fields x and y we have </li></ul><ul><li> x Y-  y X =  X,Y  </li></ul><ul><li>Where  X,Y  is the Lie bracket of vector fields X and Y. </li></ul>
    134. 134. Weitzenback Connections ( JABAS) <ul><li>Make the spin connection , non-zero torsion but zero curvature form leads to weitzenback connection. </li></ul><ul><li>Zero curvature means that there is local moving frame provided the spacetime is simply connected, since the parallel transport of the tetrad is path independent .There is a global moving frame provided the space time is a parallelizable manifold. </li></ul>
    135. 135. L-theory, Lie Bracket ( JAMBU) <ul><li>1. Derivative of f along the vector field X. </li></ul><ul><li>2. Einstein Summation Convention 1-form. </li></ul><ul><li>3. Inner product. </li></ul><ul><li>4. Isometry. </li></ul><ul><li>5. Killing Field. </li></ul><ul><li>6. Global Isometry. </li></ul><ul><li>7. Linear Isometry. </li></ul><ul><li>8. Spin Connection </li></ul>
    136. 136. Derivative Of F Along The Vector Field X ( SWARNAPRASTHA) <ul><li>Given a function f:M  R and a vector field X defined on M. </li></ul><ul><li>One defines Lie derivative of f at point p  M as </li></ul><ul><li>£ x f (p) =X p (f) =  x f (p) the usual derivative of f along the vector field X. </li></ul>
    137. 137. Einstein Summation Convention 1-form( CHANDRASHEKHAR) <ul><li>£ x f(p)= d f(p)  X (P)  </li></ul><ul><li>df = differential of f. </li></ul><ul><li>Df : M  T * M is the 1- form. </li></ul>
    138. 138. Inner Product (ABARTAN) <ul><li>df(p)  X(P)  </li></ul><ul><li>Inner product of differential of f ( at point p in M) being taken w.r.t the vector field X ( a point p). </li></ul>
    139. 139. Isometric ( MANDARHAHIN) <ul><li>Distance preserving isomorphism between metric space. </li></ul><ul><li>Let X and Y be metric spaces with metric dy and dx . </li></ul><ul><li>A map f: X  Y is called distance preserving if for any x , y  X one has dy( f ( x), f ( y)) = d x (x ,) </li></ul>
    140. 140. Killing Field (PANCHJANN) <ul><li>A vector field X is a killing field if the Lie derivative w.r.t X of the metric g vanish </li></ul><ul><li>£ x g = 0 </li></ul>
    141. 141. Global Isometric ( SINGHAL) <ul><li>Bijective distance preserving maps. </li></ul><ul><li>Any reflection, translation, rotation is a global isometry on Eucledean space. </li></ul>
    142. 142. Linear Isometric ( LANKA) <ul><li>Given two normed vector spaces V and W a linear isometric is linear map. </li></ul><ul><li>f :V  W that preserves the nom </li></ul><ul><li>||f(v)||=||v|| for all v in V. </li></ul>
    143. 143. Spin Connection (RAMNAK) <ul><li>Spin connection is a connection on a spin or bundle </li></ul>
    144. 144. Descrete-1 (PUSKAR) <ul><li>1.CONVERGENCE SPACE </li></ul><ul><li>2.UNIFORM SPACE </li></ul>
    146. 146. UNIFORM SPACE (RAMNAK)
    147. 147. Descrete-2 <ul><li>1.Classica Module Space </li></ul><ul><li>2.Quantum Module Space </li></ul>
    150. 150. Descrete-3 <ul><li>1.Product Space </li></ul><ul><li>2.Compact Space </li></ul><ul><li>3.Locally Compact Space </li></ul>
    151. 151. Descrete-4 (AGNIDH) <ul><li>1.Satadhenu </li></ul><ul><li>2.Swayangbhoj </li></ul><ul><li>3.Hyperbola Action </li></ul><ul><li>4.Hyperbollic Space </li></ul>
    152. 152. (SATADHENU)
    153. 153. ( SWAYANGBHOJ)
    154. 154. Hyperbolic Motion ( KRITOBARMA) <ul><li>In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in model. </li></ul><ul><li>Upper half-plane model </li></ul><ul><li>HP=  X,Y  :Y>0 </li></ul>
    155. 155. Hyperbolic Geometry Space ( BIDURATH) <ul><li>Poincare ½-Plane Model. </li></ul><ul><li>In non-Euclidean Geometry , the Poincare ½ plane model is the upper half, together with a metric , the Poincare metric , that makes it a model of two-dimensional hyperbolic model. </li></ul>
    156. 156. Base States (BHAJAMAN) <ul><li>1. Path of spin-1 </li></ul><ul><li>2. Path of spin-1A </li></ul><ul><li>3. Path of spin-1B </li></ul><ul><li>4. Path of spin-1C </li></ul><ul><li>5. Filters in series </li></ul><ul><li>6. Filters in series-A </li></ul><ul><li>7. 3- Filters in series </li></ul>
    157. 157. Path Of Spin – 1 ( SATADHENU) <ul><li>Stern-Gerlach Apparatus </li></ul>
    158. 158. Path Of Spin-1 A ( KRITABARMA)
    159. 159. Path Of Spin-1b ( KRITABARMA)
    160. 160. Path Of Spin -1 C ( SWANGBHOJ)
    161. 161. FILTERS IN Series (SUR)
    162. 162. Filters In Series-a (BIDURATH)
    163. 163. 3-filters In Series (SINI)
    164. 164. M-theory <ul><li>11 th dimension ( ROW & COLUMN). </li></ul><ul><li>Not a part of moving frames </li></ul>
    165. 165. M-theory (ROW) <ul><li>1.Type-i String </li></ul><ul><li>2.Type-iia String </li></ul><ul><li>3.Type-ii B String </li></ul><ul><li>4.String With Loose Ends </li></ul><ul><li>5. String With Loose Ends </li></ul><ul><li>6. Nambu 3-aljebra( Lie 3-aljebra) </li></ul><ul><li>7.Non-associative Generalisation Of Lie Aljebra </li></ul><ul><li>8. Weakly Interacting </li></ul><ul><li>9.Strongly Interacting </li></ul><ul><li>10.Prosen </li></ul><ul><li>11. Anu </li></ul>
    166. 166. Chiral (NIMLOCHI) <ul><li>A phenomena is said to be chiral if it is not identical to its mirror image. </li></ul><ul><li>Symmetry translation between the two is called parity . </li></ul><ul><li>Chirality is same as helicity. </li></ul>
    167. 167. NAMBU 3-ALJEBRA (NIMLOCHI)
    168. 168. Non-associative Generalization Of Lie Algebra (BRISNI) <ul><li>A binary operation on a set S does not satisfy the associative law is called non-associative. </li></ul><ul><li>(XxY)xY  Xx(YxZ) </li></ul><ul><li>For any X,Y,Z  S </li></ul>
    169. 169. Weak Interactions (KINGKIN) <ul><li>It is due to the exchange of heavy W and Z bosons. </li></ul><ul><li>Most familiar is beta decay and the associated radio activity. </li></ul>
    170. 170. Strongly Interacting (SINI) <ul><li>Strong interaction holds quarks and gluons together to form protons and neutrons. </li></ul>
    171. 171. Type-I String (AYUJIT) <ul><li>Is one of the five consistent super symmetric string theories in 10-dimensions It is the only one whose strings are unorientable and which contains only closed stings and also open strings. </li></ul>
    172. 172. Type-IIA String (SATAJIT) <ul><li>At low energies Type IIA string theory is described by type IIa super gravity in 10 dimensions which is a non-chiral theory (1,1) d=10 super gravity. </li></ul><ul><li>The fact that the anomalies in this theory does not cancel is therefore trivial. </li></ul><ul><li>Belong to simplistic topology and algebraic geometry particularly Groove-Witten invariant. </li></ul>
    173. 173. Type –IIB String (SAHASRAJIT) <ul><li>At low energies Type IIB string theory is described by type IIB super gravity in 10 dimensions which is a chiral theory (2,0) d=10 super gravity. </li></ul><ul><li>The fact that the anomalies in this theory cancel is therefore non- trivial. </li></ul><ul><li>Belong to algebraic geometry specially the deformation theory of complex structure originally studied by Kunilike Kodaire & Doland C Spencer. </li></ul>
    174. 174. String With Closed End (KAPATOROM) <ul><li>Strings with closed loops are free to move from membrane to membrane. </li></ul>
    175. 175. String With Loose Ends (BILOM) <ul><li>End points of the strings would not be able to move with complete freedom as they were attached or struck within certain region of space. </li></ul>
    176. 176. E8 (PRASEN) <ul><li>E8 is the name of a family of closed related structures. </li></ul><ul><li>It is the name of some exceptional simple Lie groups. </li></ul><ul><li>E8 has rank 8 and dimension 248. </li></ul>
    177. 177. U-duality (ANU) <ul><li>U-duality is a symmetry of sting theory or M-theory combining S-duality and T-duality transformation. </li></ul>
    178. 178. M-theory(COLUMN) <ul><li>1.T-duality </li></ul><ul><li>2.S-duality </li></ul><ul><li>3.Supergravity </li></ul><ul><li>4.Dimensional Reduction To A Line </li></ul><ul><li>5.Dimensional Reduction To A Circle </li></ul><ul><li>6.Heteriotic String( E8xe8 + So(32) </li></ul><ul><li>7.Heteriotic So(32) </li></ul><ul><li>8.Joy </li></ul><ul><li>9.Judhajit </li></ul><ul><li>10.Juldhan </li></ul>
    179. 179. Anamoly (ANAMITRA) <ul><li>An anomaly is an irregularity or a mis proportion on something that is strange or unusual or unique. </li></ul><ul><li>A chiral anomaly is the anomalous non-conservation of a chiral current. </li></ul>
    180. 180. S& T-duality (SUMITRA) <ul><li>S&T – duality is an equivalence of two quantum fields, string theory or M-theory. </li></ul><ul><li>S-duality transforms maps the states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory. </li></ul><ul><li>T-duality transformation the radius R of that direction will be changed to 1/R and wrapped string states will be exchanged with high-momentum string states in the dual theory. </li></ul>
    181. 181. Super gravity (KANGBAL) <ul><li>Super gravity is a field theory that combines the principle of super symmetry and general relativity. </li></ul>
    182. 182. Anthropic Explanation (KONI) <ul><li>As string theory presently understood it appears to contain a large number of distance meta-stable vacua perhaps 10 500 or more. </li></ul><ul><li>Each of these corresponds to a different possible universe with a different collection of particles and forces. </li></ul>
    183. 183. Dimensional Reduction To A Circle ( KONI) <ul><li>Get Type IIA String theory </li></ul>
    184. 184. Holographic Principle ( IBPHALAK) <ul><li>Holographic Principle Which States That The Description Of The Oscillations Of The Surface Of A Black hole Must Also Describe The Space Time Around It. </li></ul>
    185. 185. Dimensional Reduction To A Line Segment ( EBOFALOK) <ul><li>Get Heterotic SO(32) String Theory </li></ul>
    187. 187. Heteriotic (E8XE8+ SO(32))(JUGDHAR) <ul><li>Heteriotic string is a peculiar mixture of bosonic string and superstring. </li></ul><ul><li>Left moving excitation think that they lie on a bosonic string propagating in D=26 dimension. </li></ul><ul><li>Right moving excitations think that they belong to a superstring in D=10 dimension. </li></ul>
    189. 189. Heteriotic So (32) (KRUR) (AKRUR) <ul><li>Orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of a matrix multiplication. </li></ul>