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  • Form  A mapping from a vector space

Connection form Connection form Presentation Transcript

  • CONNECTION FORM -MOVING FRAME( E=mc 2 ) -DIFFERENTIAL FORM
  • A connection form is a manner of organizing data of a connection language of moving frame and differential form
  • Moving Frame
    • 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.
  • Differential Form
    • Differential form is a mathematical concept in fields of a multivariative calculus differential topology and tensors.
  • Moving Frame
    • 1.PROJECTIVE FRAME
    • 2.AFFINE FRAME
    • 3.LINEAR FRAME
    • 4. ORTHONORMAL FRAME
    • 5. EUCLIDEAN FRAME
    • 6. CO-FRAME
    • 7. KLEIN GEO-FRAME
    • 8. FRENET-SERRET FRAME
    • 9.DARBOAX FRAME
    • 10.DESCRETE FRAME/K-THEORY
    • 11.TIME
  • Projective Frame
    • 1. Sierpinki Space / Homology
    • 2. Pre-regular Space/ Cross Ratio
    • 3. Second Countable Space /Quadrangle Theorem
    • 4. Normal Space /Harmonic Range
    • 5. Lindelof Space/Paskals Theorem
    • 6. Functional Space/ Pappus Theorem
    • 7. Fully Normal Space / Brain Chon
    • 8. Counter Space
  • Pre–regular Space (SUNDARANAND)
    • X is a pre-regular space if any two topological distinguasible points can be separated by neighborhood.
  • Homology ( SUNDARANAND)
    • 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.
  • Functional Space ( SARBANAND)
    • Functional space is A set of A given kind from A set X to A set Y.
  • Cross Ratio ( SARBANANDA)
    • Cross ratio of four points is only numerical invariant of projective geometry.
  • Second Countable Space ( CHANDRASEKHAR)
    • Satisfy the second axiom of count ability.
    • Has a comfortable base.
    • Is separable and lindelof.
  • Quadrangle Theorem (Chandrashekhar)
    • If two quadrangles have 5 pairs of corresponding sides meeting in collinear points the sixth pair meets on the same line.
  • Normed Space (AMBER)
    • CONSIST OF T4,T5 &T6
  • Harmonic Range ( AMBER)
    • Construction of two pairs of points harmonically have cross ratio 1.
  • Lindelof Space (KAPILAMBER)
    • Is a topological space in which every open cover has a countable sub cover.
    • More commonly used notion of compactness.
    • Strongly lindof / suslin.
  • Pascal's Theorem ( KAPILAMBAR)
  • Fully Normal Space ( VAB)
    • Is a topological space in which every open cover admits an open locally finite refinement.
  • Pappus’s Theorem ( VAB)
  • Sierpinki Space (BAMAN)
    • It is a smallest example of a topological space which is neither trivial nor discrete.
    • Is a finite topological space with two points, only one of which is closed.
  • Brain Chon (BAMAN)
  • Counter Space (BHIRUK)
  • Affine Frame
    • An affine space is a set with a faithful freely transitive vector space action i,e a tensor for the vector space
    •  :SXS  (a,b)|  (a,b)
  • Affine Frame
    • 1. QUOTENT SPACE / David Kay’s Axiom
    • 2. FRECHET SPACE (T1) / David Kay’s Axiom ( AS1)
    • 3. HOUSDORF SPACE(T2) / David Kay’s Axiom ( AS2)
    • 4. REGULAR SPACE(T3) / David Kay’s Axiom ( AS3)
    • 5. NORMAL SPACE(T4) / David Kay’s Axiom ( AS4)
    • 6. NORMAL SPACE(T5) / David Kay’s Axiom ( AS5)
    • 7. NORMAL SPACE(T6) / David Kay’s Axiom ( AS6)
  • Quotient Space (BHUTESH)
    • Identification space is intuitively speaking the result of identifying or “gluing together” certain points of a given space.
    • Equivalence relation
  • David Kay’s Axim (BHUTESH)
    • David Kay’s description of 3-dimensional affine space is as follows….
    • “ An affine space is any system of points ,lines and planes which satisfy 6 axioms.”
  • T1 Space, Frechet Space (SANGBART)
    • It is complete as a uniform space.
    • It is locally convex.
  • ASI (SANGBARTA)
    • Two distinct points determine a unique line.
  • T2 Space, Hausdorff Space (BIKRITAKH)
    • A topological space in which points can be separated by neighborhood.
  • AS2 (BIKRITAKH)
    • Three non-linear points determine a unique plane.
  • T3 Space (SANGHAR)
    • Normal space
  • AS3 ( SANGHAR)
    • If two points lie in a plane then the line determined by these points lies in that plane.
  • T4 Space (DANDAPANI)
    • Normal space
  • AS4 (DANDAPANI)
    • If two planes meet their intersection is a line.
  • T5 space (Chakrapani)
    • Normal space
  • AS5 ( CHAKRAPANI)
    • There exist at least four non-linear points and at least one plane.
    • Each plain contains at least three non-linear points.
  • T6 Space (Kal)
    • Normal space
  • AS6 ( KAL)
    • Given any two non co-planer lines, there exist a unique plane through the first line which is parallel to second line.
  • Linear Frame
    • 1. CO-ODINATE SPACE.
    • 2. SYMMETRIC SPACE/ Standard Basis.
    • 3. CONFORMAL SPACE /Hamel Basis.
    • 4. DUAL SPACE.
    • 5. METRIC SPACE /Orthonormal Base.
    • 6. LUSINS SPACE /Schander Base.
    • 7. ALJEBIC DUAL SPACE.
    • 8. ANTI DE-SITTER SPACE.
  • Co-ordinate Space ( Kamadiswar)
    • Co- Ordinate space is a proptotypical example of n-dimensional vector space over a field F.
  • Symmetric Space (AMRITAKH)
    • Same as R0 space
  • Standard Basis ( Amritakh)
    • Standard basis is a sequence of or the unit vector.
    • Standard basis of a n-dimensional Euclidean space R n is the basis obtained by taking the n-basis vector.
    •  e i : 1  I  n 
    • Where e i is the vector with a 1 in the co-ordinate and 0 elsewhere.
  • Algebraic Dual Space (KAPALI)
    • 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.
  • Conformal Space (BOMKESH)
    • Conformal geometry is the study of the set of angle-preserving (conformal) transformation on a Riemann manifold.
  • Hamel Basis (BOMKESH)
    • Where the number of tensors in the linear combinations
    • A 1 v 1 +……..+a n v n
    • Is always finite.
  • Metric Space (TRIPURESH)
    • Metric space is a set where a notion of distance (metric) between elements of the set is defined.
  • Orthonormal Basis (TRIPURESH)
    • Orthonormal basis of an inner product space V is a set of basis vectors whose elements are mutually orthonormal and of magnitude 1.
  • Dual Space (ESWAR)
    • Defined a finite-dimensional vector space can be used for defined tensors which are studied in tensor space,
  • Dual Space (ESHAWAR)
    • 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.
  • Lusin Space (NAKULISH)
    • Lusin space is topological space such that some weaken topology makes it into a Polish space.
  • Schander Basis (NAKULESH)
    • 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.
  • Anti De-sitter Space (KHIROKANTHHA)
    • Anti de-sitter space can be visualized as the lorentizian analogs of a sphere in a space of an additional dimension.
  • Orthonormal Frame
    • 1. FORCK SPACE / Multilinear Operation.
    • 2. LP space / Bilinear Operation.
    • 3. ORBIT SPACE / Sesquilinear Form.
  • Fock Space (UNMATT)
    • Fork space is a algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particle.
    • Fork space is a Hilbert space made from direct sum of tensor product of single-particle Hilbert space.
  • Multilinear Operation (Unmatt)
    • Multilinear operation is a map of type
    • f : V n  k
    • Where V is a vector space over field k, that is separate linear in each its N variable.
  • Lebesgue Space (LP)( LAMBAKARNA)
    • P-form can be extended to vectors having an infinite number of components; yielding the space Lp.
  • Bilinear Operation ( Lambakarna)
    • 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.
    • 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.
  • Orbit Space (BISHES)
    • 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 .
    • The quotient space under this relation is called the Orbit space.
  • Sesquilinear Form ( BISHES)
    • A Sesquilinear form on a complex vector space v is a map in one argument and anti-linear in other.
    • If  (x+y, z+w)
    • =  (x, y)+  (x, w)+  (y, z)+  (y, w)
    •  (ax, by)= ãb  (x, y)
    • For all x, y, z,w in V AND a ,b in C
  • Euclidean Frame
    • 1. Baire Space
    • 2.Topological Space.
    • 3. Tychonoff Space.
  • T 3 ½ , Tychonoff Space (Trisangkeswar)
    • Completely regular space.
  • Topological Space ( JAGANATH)
    • Topological spaces are mathematical structures that allow the formal definition of concepts:-
      • Convergence
      • Connectedness
      • Continuity
  • Baire Space ( SAMBRANAND)
    • Baire space is a very large and “enough” points for certain limit process of closed set with empty interior.
  • Co-frame
    • 1.SUSLIN SPACE/Dual Frame
    • 2.SCHWART SPACE/Co-tangent Bundle
    • 3.RANDOM SPACE/Kronker Delta
  • Suslin Space ( NIRMISH)
    • Is a separable completely amortizable topological space.
    • I, e a space homogeneous to a complete metric space that has a comfortable dense subset.
    • A Suslin space is the image of a polish space under a continuous mapping.
  • Dual Frame (NIRMISH)
    • A moving frame determines a dual frame of co-tangent bundle over U, which is sometimes also called Moving Frame.
    • This is a n-tuple of smooth1-form a 1 ,a 2 ,--a n which are linearly independent at each point q in U.
  • Radon Space (NANDIKESWAR)
    • A Random space is a topological space such that every finite Boral measure is inner regular.
  • Co-tangent Bundle (NANDIKESHAR)
    • Co-tangent bundle of a smooth manifold is the vector bundle of all the co-tangent space at every point in the manifold.
  • Schwartz Space (KRODHISH)
    • Schwartz space is the function space of rapidly decreasing function.
  • Kroncker Delta ( KRODHISH)
    • Kroncker delta is a function of two bundles which is 1 if they are equal, 0 if otherwise.
  • Darboux Frame
    • 1.CONTOUR SPACE/Adapted Frame
    • 2.LOCALLY CONVEX SPACE/ Maurer – Cartan Frame
    • 3.F SPACE /Push Forward & Pull Back
  • Contour Space (ABHIRUK)
    • Each lip contour is a point in an 80-dimensional “ contour space”.
  • Push Backward & Forward (ABHIRUK)
    • Let  :M  N be a smooth map of smooth manifolds.
    • Given some x  M, the Push forward of  at x is a linear map
    • d  x :T x M  T  (x) N from tangent space of M at x to the tangent space of N at  (x).
    • The applicant vector X is sometimes called the Push Forward of x by  .
  • Locally Convex Space ( BAKRANATH)
    • Locally convex space is defined either in terms of convex set or equivalent in terms of semi norm.
  • Adapted Frame (BAKRANATH)
    • Let  :M  E n be an embedding of p-dimensional smooth manifold into a euclidean space.
    • 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).
  • F-space (RAKHASESWAR)
    • Is a vector space V over the real or complex number together with a metric
    • D:VXV  R
    • V is continuous transition invariant.
  • Maurer-cartan Form (RAKHASHESWAR)
    • Let g = TeG be the tangent space of a Lie group G at identity G acts on itself by left translation
    • L:GXG  G such that g  G
    • We have Lg:G  G where Lg(h)=gh this induces a map of tangent bundle on itself (Lg)*Thg  TghG
    • Aleft invariant vector field is a section x of TG such that
    • (Lg)*X=X  g  G
    • The Mauer-Cartan form w is g-valued one-form on G defined on vectors v  TgG by formula
    • w(v)=(Lg -1 )*v
  • Klein Geometry
    • Accessible Space
    • 2. Sequential Space
    • 3. De-sitter Space
    • 4. Homogeneous Space
    • 5. Symmetric Space
    • 6. R0 Space
  • Sequential Space (VADRASEN)
    • Sequential space /Frechet Urysohm space satisfy a very weak axiom of compatibility.
    • A sequential space is a space X satisfying one of the following equivalent conditions.
    • 1. Every sequential open subset of X is open.
    • 2. Every sequential closed subset of X is closed.
  • De-sitter Space (BHIMLOCHAN)
    • De-Sitter space is Lorentz an analogue of an n-sphere (with its canonical Riemannian metric).
    • De-Sitter space can be defined as a sub manifold of Murkowski space in one higher
  • Homogeneous Space (BAKRAKUNDA)
    • Traditional spaces are homogeneous space, but not for a uniquely determined group.
    • Changing the group changes the appropriate language.
  • Regular Space (BAIDYANATH)
    • X is a T3 space if and only if it is both regular and Hausdorff.
  • R 0 Space (RURU)
    • Let X be a topological space and let x and y be points in X .
    • 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.
  • Accessible Space ( ASITANGA)
    • X is a T1 space if any two distinct points in X can be separated.
  • K-theory
    • 1.BANACH SPACE/ Representative K-theory
    • 2.ADJUNCTION SPACE/Disjoint Union
  • Adjunction Space
    • 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.
    • The resulting quotient space is denoted X/A.
  • DISJOINT UNION ( JOGISH)
  • Banach Space (KAPALI)
    • Banach space is a vector space V over the real or complex number with a nom !!.!!.
  • REPRESENTATIVE K-THEORY ( TRAMBAK)
  • Franet-serret Frame
    • 1.Affine Space
    • 2.Twistor Space
    • 3.Cauchy Space
    • 4.Euclidean Space
  • Affine Space (SHUR)
    • An affine space is any system of points , lines and planes which satisfy 6 axioms.
  • Twister Space (DEBBAHU)
    • Is a 4-dimensional complex space
    • T:=C 4
    • It has associated to it the double fibration of flag manifold
    • P   F   M
    • Where projective Twister space
    • P:=F1(T)=P3(C)=P( C4)
  • Cauchy Space (SHIV)
    • A Cauchy space is a set X and a collection of C of proper filters in power set P(X).
  • Euclidean Space (MAHADAR)
    • An n-dimensional space with a notion of distance and angle that obey Euclidean relationship.
  • Differential Form
    • Connection Form
    • 2. Curvature Form
    • 3. Torsion Form
    • 4. Solder Form
    • 5. Levi-Civita Connection Form
    • Lie Bracket Form
    • Descrete-1
    • 8. Descrete-2
    • 9. Descrete-3
    • 10. Descrete-4
    • 11. Base States
  • Connection Form (Shak)
    • 1. Principal Connection
    • 2. Compatible Connection
    • 3. Cartan Connection
    • 4. Ehesmann Connection
    • 5. Associated to Principal
    • 6. Global Connection Form
    • 7. Affine Connection
  • Principal Connection ( PURAJAB)
    • Let  U  be an open cover of M, along with G-frame on each U, denoted by e U .
    • 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.
  • Compatible Connection (PABMAN)
    • 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.
  • Cretan Connection (DHRUBHANIK)
    • Cretan Connection are closely related to pseudo-group structure on a manifold.
    • Pseudo group is an extension of group concept , but one that grew out of the geometric approach of Sophus Lie.
  • Ehesmann Connection ( CHITRAREK)
    • Ehesmann connection is a version of the notion of a connection which is defined on ordinary fiber bundle.
  • Associated To Principle ( BAHURUP)
    • A Principle G-connection w in a principal G-bundle P  M gives rise to a collection of connection focus on M.
  • Global Connection Form ( BISHWAVAB)
    • 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.
  • Affine Connection (MANAJAB)
    • 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.
  • Curvature Form (KUSH)
    • 1.Curvilinear Co-ordinate
    • 2.Non-tensor
    • 3.Parallel Transport
    • 4.Geodesics
    • 5.Co-variant Derivative
    • 6.Curvature Tensor
    • 7. Bianchi Identity
  • Curvilinear Co-ordinate (BASU)
    • Curvilinear co-ordinates are co-ordinate in which angles between axes can changes from point to point.
  • Nontensors ( HARRUCHI)
    • Nontensor is a tensor like quantity N  that behaves like a tensor in the raising and lowering of indices
    • N  = g  N  N  = g  N 
    • But does not transform like a tensor under a co-ordinate transformation.
  • Parallel Transport ( NAVIGUPTA)
  • Geodesics (SUTABRATA)
    • Suppose we have a point x u that moves along a truck in physical space time.
    • Suppose the track is parameterized with the quantity  .The variation of velocity upon parallel displacement along the track can be calculated.
    • If there are no force acting on the point then the velocity is unchanged and we have Geodesics Equation.
  • Co-variant Derivative ( BIKIKAKT)
    • The partial derivative of a vector w.r.t a space-time co-ordinate is composed of two parts.
    • Normal partial derivative minus the change in the vector due to parallel transport.
    • A  :  =A  1  _A a T  a
  • Curvature Tensor (BAMDEV)
    • The curvature K of a surface is simply the angle through a vector is turned as we take it around an infinitesimal closed path.
  • Bianchi Identity (BASUDAN)
    • Following differential relation known as Bianchi identity
    • R   :  +R   :  +R   :  =0
  • Torsion Form ( KOUNCH)
    • 1. Curvature Tensor
    • 2. Cyclic Sum
    • 3. Curvature Form
    • 4. Affinity Parameterized Geodesic
    • 5. Twisting of Reference Frame
    • 6. Torsion of a Filament
    • 7. Affine Developments
  • Curvature Tensor (AM)
    • Curvature tensor (  ) is a mapping
    • TM  TM  End (TM)
    • Defined on vector field x, y & z
    • R(X,Y)Z=  x  y Z-  y  x Z-   x, y  Z
  • Cyclic Sum ( MADHUBRAHA)
    •  (R(X,Y)X):=
    • R(X,Y)Z+R(Y,Z)X+R(Z,X)Y
  • Curvature Form (MEGHAPRISTA)
    • Curvature form is the gl(n)-valued 2-form.
    •  =Dw = dw +w  w
    • D denotes the exterior covariant derivative
    • 1. D  = 
    • 2. D  =0
  • Affinely Parametrized Geodesic ( SUDHAMA)
    • Suppose y(t) is a curve on M
    • Then  y (t) Y (t)=0
    • For all time t is the domain of Y.
  • Twisting Of Reference Frame ( BHAJISTA)
    • Frenet-Serret Formula describe how a particular moving frame twists along a curve.
  • Torsion Of A Filament ( LOHITAN)
    • Length –maximizing (geodesic) configuration and its energy-minimizing configuration.
  • Affine Developments (BANASPATY)
    • 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
    • Ċ t =  t 0 x t , c 0 =0
    • Where  t 0 :Tx t M  Tx 0 M
    • is the parallel transport associated to  .
  • Solder Form (SHALMLI)
    • 1. Smooth Manifold
    • 2. Symplectic Form
    • 3. Complex Manifold
    • 4. Vierbein or Tetrad Theory
    • 5. Signature
    • 6. Palatini Action
    • 7. Interwiners
  • Smooth Manifold ( SURACHAN)
    • A differentiable manifold for which all the transitions maps are smooth.
    • C 
  • Symplectic Form ( SOUMANAS)
    • A symplectic form on a manifold M is a non-degenerative closed two form w.
  • Complex Manifold ( RAMANAK)
    • Complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space.
  • Vierbeib Or Tetrad Theory ( DEVBARS)
    • Special case of application of Cretan connection in four – dimensional manifold.
  • Signature (PARIVADRA)
    • Signature is the number of positive and negative Eigen values of the vector,
  • Palatini Action ( APAYAN)
    • 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  .
  • Interwiners (ABIGAN)
    • 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.
    • Thus an interwiner is an equivalent map in the special case of two linear representations/ action.
  • Levi-civita Connection (PLAKH)
    • 1. Preserve the metric
    • 2. Derivative along curve
    • 3. Unit sphere in R3
    • 4. Isomorphism
    • 5. Leibniz Rule
    • 6. Torsion Free
    • 7. Weitzenbock Connection
  • Preserve The Metric ( SHIV)
    • Preserve the metric I,e for any vector field x, y, z we have
    • X (g (Y,Z))=g (  x Y, Z) + g (Y,  x Z)
    • Where X (g (Y,Z)) denotes derivative of a function g (Y,Z) along the vector field X.
  • Derivative Along Curve (SUVADRA)
    • Derivative along curve D = Pullback
    • Given a smooth curve Y on (M, g) and a vector field V along Y its derivative determined by
    • D t V =  ý (t) V
  • Unit Sphere In R 3 ( SHANT)
    • 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.
  • Isomorphism (KHEM)
    • Parallel transport along a curve w.r.t a connection define isomorphism between the tangent space at that point of the curve.
  • Leibniz RulE (AMRIT)
    • C  (S 2 ) linear is first variable.
    • Consider a map f:S 2  R m  Y ( m ), m 
    • The map f is constant hence differential vanishes
    • dm f (x)=  dm Y (x) m  +  Y (m) ,X (m)  =0
    •  (  x Y) (m), m  =0
  • Torsion Free ( ABHAY)
    • Torsion free i,e for any vector fields x and y we have
    •  x Y-  y X =  X,Y 
    • Where  X,Y  is the Lie bracket of vector fields X and Y.
  • Weitzenback Connections ( JABAS)
    • Make the spin connection , non-zero torsion but zero curvature form leads to weitzenback connection.
    • 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.
  • L-theory, Lie Bracket ( JAMBU)
    • 1. Derivative of f along the vector field X.
    • 2. Einstein Summation Convention 1-form.
    • 3. Inner product.
    • 4. Isometry.
    • 5. Killing Field.
    • 6. Global Isometry.
    • 7. Linear Isometry.
    • 8. Spin Connection
  • Derivative Of F Along The Vector Field X ( SWARNAPRASTHA)
    • Given a function f:M  R and a vector field X defined on M.
    • One defines Lie derivative of f at point p  M as
    • £ x f (p) =X p (f) =  x f (p) the usual derivative of f along the vector field X.
  • Einstein Summation Convention 1-form( CHANDRASHEKHAR)
    • £ x f(p)= d f(p)  X (P) 
    • df = differential of f.
    • Df : M  T * M is the 1- form.
  • Inner Product (ABARTAN)
    • df(p)  X(P) 
    • Inner product of differential of f ( at point p in M) being taken w.r.t the vector field X ( a point p).
  • Isometric ( MANDARHAHIN)
    • Distance preserving isomorphism between metric space.
    • Let X and Y be metric spaces with metric dy and dx .
    • 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 ,)
  • Killing Field (PANCHJANN)
    • A vector field X is a killing field if the Lie derivative w.r.t X of the metric g vanish
    • £ x g = 0
  • Global Isometric ( SINGHAL)
    • Bijective distance preserving maps.
    • Any reflection, translation, rotation is a global isometry on Eucledean space.
  • Linear Isometric ( LANKA)
    • Given two normed vector spaces V and W a linear isometric is linear map.
    • f :V  W that preserves the nom
    • ||f(v)||=||v|| for all v in V.
  • Spin Connection (RAMNAK)
    • Spin connection is a connection on a spin or bundle
  • Descrete-1 (PUSKAR)
    • 1.CONVERGENCE SPACE
    • 2.UNIFORM SPACE
  • CONVERGENCE SPACE (GHATAK)
  • UNIFORM SPACE (RAMNAK)
  • Descrete-2
    • 1.Classica Module Space
    • 2.Quantum Module Space
  • CLASSICAL MODULLI SPACE (STHANU)
  • QUANTUM MODULLI SPACE(SARBANANDA)
  • Descrete-3
    • 1.Product Space
    • 2.Compact Space
    • 3.Locally Compact Space
  • Descrete-4 (AGNIDH)
    • 1.Satadhenu
    • 2.Swayangbhoj
    • 3.Hyperbola Action
    • 4.Hyperbollic Space
  • (SATADHENU)
  • ( SWAYANGBHOJ)
  • Hyperbolic Motion ( KRITOBARMA)
    • In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in model.
    • Upper half-plane model
    • HP=  X,Y  :Y>0
  • Hyperbolic Geometry Space ( BIDURATH)
    • Poincare ½-Plane Model.
    • 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.
  • Base States (BHAJAMAN)
    • 1. Path of spin-1
    • 2. Path of spin-1A
    • 3. Path of spin-1B
    • 4. Path of spin-1C
    • 5. Filters in series
    • 6. Filters in series-A
    • 7. 3- Filters in series
  • Path Of Spin – 1 ( SATADHENU)
    • Stern-Gerlach Apparatus
  • Path Of Spin-1 A ( KRITABARMA)
  • Path Of Spin-1b ( KRITABARMA)
  • Path Of Spin -1 C ( SWANGBHOJ)
  • FILTERS IN Series (SUR)
  • Filters In Series-a (BIDURATH)
  • 3-filters In Series (SINI)
  • M-theory
    • 11 th dimension ( ROW & COLUMN).
    • Not a part of moving frames
  • M-theory (ROW)
    • 1.Type-i String
    • 2.Type-iia String
    • 3.Type-ii B String
    • 4.String With Loose Ends
    • 5. String With Loose Ends
    • 6. Nambu 3-aljebra( Lie 3-aljebra)
    • 7.Non-associative Generalisation Of Lie Aljebra
    • 8. Weakly Interacting
    • 9.Strongly Interacting
    • 10.Prosen
    • 11. Anu
  • Chiral (NIMLOCHI)
    • A phenomena is said to be chiral if it is not identical to its mirror image.
    • Symmetry translation between the two is called parity .
    • Chirality is same as helicity.
  • NAMBU 3-ALJEBRA (NIMLOCHI)
  • Non-associative Generalization Of Lie Algebra (BRISNI)
    • A binary operation on a set S does not satisfy the associative law is called non-associative.
    • (XxY)xY  Xx(YxZ)
    • For any X,Y,Z  S
  • Weak Interactions (KINGKIN)
    • It is due to the exchange of heavy W and Z bosons.
    • Most familiar is beta decay and the associated radio activity.
  • Strongly Interacting (SINI)
    • Strong interaction holds quarks and gluons together to form protons and neutrons.
  • Type-I String (AYUJIT)
    • 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.
  • Type-IIA String (SATAJIT)
    • 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.
    • The fact that the anomalies in this theory does not cancel is therefore trivial.
    • Belong to simplistic topology and algebraic geometry particularly Groove-Witten invariant.
  • Type –IIB String (SAHASRAJIT)
    • 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.
    • The fact that the anomalies in this theory cancel is therefore non- trivial.
    • Belong to algebraic geometry specially the deformation theory of complex structure originally studied by Kunilike Kodaire & Doland C Spencer.
  • String With Closed End (KAPATOROM)
    • Strings with closed loops are free to move from membrane to membrane.
  • String With Loose Ends (BILOM)
    • 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.
  • E8 (PRASEN)
    • E8 is the name of a family of closed related structures.
    • It is the name of some exceptional simple Lie groups.
    • E8 has rank 8 and dimension 248.
  • U-duality (ANU)
    • U-duality is a symmetry of sting theory or M-theory combining S-duality and T-duality transformation.
  • M-theory(COLUMN)
    • 1.T-duality
    • 2.S-duality
    • 3.Supergravity
    • 4.Dimensional Reduction To A Line
    • 5.Dimensional Reduction To A Circle
    • 6.Heteriotic String( E8xe8 + So(32)
    • 7.Heteriotic So(32)
    • 8.Joy
    • 9.Judhajit
    • 10.Juldhan
  • Anamoly (ANAMITRA)
    • An anomaly is an irregularity or a mis proportion on something that is strange or unusual or unique.
    • A chiral anomaly is the anomalous non-conservation of a chiral current.
  • S& T-duality (SUMITRA)
    • S&T – duality is an equivalence of two quantum fields, string theory or M-theory.
    • 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.
    • 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.
  • Super gravity (KANGBAL)
    • Super gravity is a field theory that combines the principle of super symmetry and general relativity.
  • Anthropic Explanation (KONI)
    • As string theory presently understood it appears to contain a large number of distance meta-stable vacua perhaps 10 500 or more.
    • Each of these corresponds to a different possible universe with a different collection of particles and forces.
  • Dimensional Reduction To A Circle ( KONI)
    • Get Type IIA String theory
  • Holographic Principle ( IBPHALAK)
    • 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.
  • Dimensional Reduction To A Line Segment ( EBOFALOK)
    • Get Heterotic SO(32) String Theory
  • TYPE IIA STRING (JUDHAJIT)
  • Heteriotic (E8XE8+ SO(32))(JUGDHAR)
    • Heteriotic string is a peculiar mixture of bosonic string and superstring.
    • Left moving excitation think that they lie on a bosonic string propagating in D=26 dimension.
    • Right moving excitations think that they belong to a superstring in D=10 dimension.
  • TYPE-IIA STRING (JUJUDHAN)
  • Heteriotic So (32) (KRUR) (AKRUR)
    • 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.