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# Connection form

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## Connection formPresentation 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
• 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.
• 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 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.
• 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
• 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.
• 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 /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
• 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).
• 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
•  (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 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 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)
• 2.Swayangbhoj
• 3.Hyperbola Action
• 4.Hyperbollic Space
• ( 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.