CONNECTION FORMCONNECTION FORM-MOVING FRAME( E=mc2)

-DIFFERENTIAL FORM

A connection form is a manner of organizing A connection form is a manner of organizing data of a connection language of moving data of a connection language of moving

frame and differential formframe and differential form

Moving FrameMoving 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 FormDifferential Form

• Differential form is a mathematical concept in fields of a multivariative calculus differential topology and tensors.

Moving FrameMoving Frame

1.PROJECTIVE FRAME2.AFFINE FRAME3.LINEAR FRAME4. ORTHONORMAL FRAME5. EUCLIDEAN FRAME 6. CO-FRAME7. KLEIN GEO-FRAME8. FRENET-SERRET FRAME9.DARBOAX FRAME10.DESCRETE FRAME/K-THEORY11.TIME

Projective FrameProjective Frame

1. Sierpinki Space /Homology

2. Pre-regular Space/ Cross Ratio3. Second Countable Space /Quadrangle

Theorem4. Normal Space /Harmonic Range5. Lindelof Space/Paskals Theorem6. Functional Space/ Pappus Theorem7. Fully Normal Space / Brain Chon8. Counter Space

Pre–regular Space (SUNDARANAND)Pre–regular Space (SUNDARANAND)

• X is a pre-regular space if any two topological distinguasible points can be separated by neighborhood.

Homology ( SUNDARANAND)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 ( SARBANAND)

• Functional space is A set of A given kind from A set X to A set Y.

Cross Ratio ( SARBANANDA)Cross Ratio ( SARBANANDA)

• Cross ratio of four points is only numerical invariant of projective geometry.

Second Countable SpaceSecond Countable Space( CHANDRASEKHAR)( CHANDRASEKHAR)

• Satisfy the second axiom of count ability.

• Has a comfortable base.• Is separable and lindelof.

Quadrangle Theorem Quadrangle Theorem (Chandrashekhar)(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)Normed Space (AMBER)

• CONSIST OF T4,T5 &T6

Harmonic Range ( AMBER)Harmonic Range ( AMBER)

• Construction of two pairs of points harmonically have cross ratio 1.

Lindelof Space (KAPILAMBER)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)Pascal's Theorem ( KAPILAMBAR)

Fully Normal Space ( VAB)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)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)Brain Chon (BAMAN)

Counter Space (BHIRUK)Counter Space (BHIRUK)

Affine FrameAffine 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 FrameAffine 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)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 David Kay’s Axim (BHUTESH)(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)T1 Space, Frechet Space (SANGBART)

• It is complete as a uniform space.• It is locally convex.

ASI (SANGBARTA)ASI (SANGBARTA)

• Two distinct points determine a unique line.

T2 Space, Hausdorff Space (BIKRITAKH)T2 Space, Hausdorff Space (BIKRITAKH)

• A topological space in which points can be separated by neighborhood.

AS2 (BIKRITAKH)AS2 (BIKRITAKH)

• Three non-linear points determine a unique plane.

T3 Space (SANGHAR)T3 Space (SANGHAR)

• Normal space

AS3 ( SANGHAR)AS3 ( SANGHAR)

• If two points lie in a plane then the line determined by these points lies in that plane.

T4 Space (DANDAPANI)T4 Space (DANDAPANI)

• Normal space

AS4 (DANDAPANI)AS4 (DANDAPANI)

• If two planes meet their intersection is a line.

T5 space (Chakrapani)T5 space (Chakrapani)

• Normal space

AS5 ( CHAKRAPANI)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)T6 Space (Kal)

• Normal space

AS6 ( KAL)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 FrameLinear 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 ( Kamadiswar)

• Co- Ordinate space is a proptotypical example of n-dimensional vector space over a field F.

Symmetric Space (AMRITAKH)Symmetric Space (AMRITAKH)

• Same as R0 space

Standard Basis ( Amritakh)Standard Basis ( Amritakh)

• Standard basis is a sequence of or the unit vector.

• Standard basis of a n-dimensional Euclidean space Rn is the basis obtained by taking the n-basis vector.

ei : 1 I n

• Where ei is the vector with a 1 in the co-ordinate and 0 elsewhere.

Algebraic Dual SpaceAlgebraic 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 Space (BOMKESH)

• Conformal geometry is the study of the set of angle-preserving (conformal) transformation on a Riemann manifold.

Hamel Basis (BOMKESH)Hamel Basis (BOMKESH)

• Where the number of tensors in the linear combinations

• A1v1+……..+anvn

• Is always finite.

Metric Space (TRIPURESH)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 (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)Dual Space (ESWAR)

• Defined a finite-dimensional vector space can be used for defined tensors which are studied in tensor space,

DualDual Space (ESHAWAR)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 (NAKULISH)

• Lusin space is topological space such that some weaken topology makes it into a Polish space.

Schander Basis (NAKULESH)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 (KHIROKANTHHA)

• Anti de-sitter space can be visualized as the lorentizian analogs of a sphere in a space of an additional dimension.

Orthonormal FrameOrthonormal Frame

• 1. FORCK SPACE / Multilinear Operation.

• 2. LP space / Bilinear Operation.

• 3. ORBIT SPACE / Sesquilinear Form.

Fock Space (UNMATT)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 (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)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 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 vםB (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)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)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 FrameEuclidean Frame

• 1. Baire Space

• 2.Topological Space.

• 3. Tychonoff Space.

T 3 T 3 ½ ,½ ,Tychonoff Space Tychonoff Space (Trisangkeswar)(Trisangkeswar)

• Completely regular space.

Topological Space ( JAGANATH)Topological Space ( JAGANATH)

• Topological spaces are mathematical structures that allow the formal definition of concepts:-

a) Convergenceb) Connectednessc) Continuity

Baire Space ( SAMBRANAND)Baire Space ( SAMBRANAND)

• Baire space is a very large and “enough” points for certain limit process of closed set with empty interior.

Co-frameCo-frame

• 1.SUSLIN SPACE/Dual Frame

• 2.SCHWART SPACE/Co-tangent Bundle

• 3.RANDOM SPACE/Kronker Delta

Suslin Space ( NIRMISH)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)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 a1,a2,--an which are linearly independent at each point q in U.

Radon Space (NANDIKESWAR)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 (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 (KRODHISH)

• Schwartz space is the function space of rapidly decreasing function.

Kroncker Delta ( KRODHISH)Kroncker Delta ( KRODHISH)

• Kroncker delta is a function of two bundles which is 1 if they are equal, 0 if otherwise.

Darboux FrameDarboux Frame

1.CONTOUR SPACE/Adapted Frame

2.LOCALLY CONVEX SPACE/ Maurer – Cartan Frame

3.F SPACE /Push Forward & Pull Back

Contour Space (ABHIRUK)Contour Space (ABHIRUK)

• Each lip contour is a point in an 80-dimensional “ contour space”.

Push Backward & Forward (ABHIRUK)Push Backward & Forward (ABHIRUK)

• Let:MN be a smooth map of smooth manifolds.

• Given some xM, the Push forward of at x is a linear map

• dx :TxMT(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 ( BAKRANATH)

• Locally convex space is defined either in terms of convex set or equivalent in terms of semi norm.

Adapted Frame (BAKRANATH)Adapted Frame (BAKRANATH)

• Let :MEn 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 (x1f1,….fn) where xM and f1 form an ortonormal basis of En such that f1….fq are tangent to (M) at (v).

F-space (RAKHASESWAR)F-space (RAKHASESWAR)

• Is a vector space V over the real or complex number together with a metric

• D:VXVR• V is continuous transition invariant.

Maurer-cartan Form (RAKHASHESWAR)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:GXGG such that g G• We have Lg:GG where Lg(h)=gh this induces a

map of tangent bundle on itself (Lg)*ThgTghG• Aleft invariant vector field is a section x of TG

such that • (Lg)*X=XgG• 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 GeometryKlein Geometry

1. Accessible Space

2. Sequential Space

3. De-sitter Space

4. Homogeneous Space

5. Symmetric Space

6. R0 Space

Sequential Space (VADRASEN)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 (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)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)Regular Space (BAIDYANATH)

• X is a T3 space if and only if it is both regular and Hausdorff.

RR00 Space (RURU) 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 R0 space if any two topologically distinguishable points in X can be separated.

Accessible Space ( ASITANGA)Accessible Space ( ASITANGA)

• X is a T1 space if any two distinct points in X can be separated.

K-theoryK-theory

1.BANACH SPACE/ Representative K-theory

2.ADJUNCTION SPACE/Disjoint Union

Adjunction SpaceAdjunction 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)DISJOINT UNION ( JOGISH)

Banach Space (KAPALI)Banach Space (KAPALI)

• Banach space is a vector space V over the real or complex number with a nom !!.!!.

REPRESENTATIVE K-THEORY REPRESENTATIVE K-THEORY ( TRAMBAK)( TRAMBAK)

Franet-serret FrameFranet-serret Frame

1.Affine Space

2.Twistor Space

3.Cauchy Space

4.Euclidean Space

Affine Space (SHUR)Affine Space (SHUR)

• An affine space is any system of points , lines and planes which satisfy 6 axioms.

Twister Space Twister Space (DEBBAHU)(DEBBAHU)

• Is a 4-dimensional complex space

• T:=C4

• 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)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)Euclidean Space (MAHADAR)

• An n-dimensional space with a notion of distance and angle that obey Euclidean relationship.

Differential FormDifferential Form

1. Connection Form2. Curvature Form3. Torsion Form4. Solder Form5. Levi-Civita Connection Form6. Lie Bracket Form7. Descrete-18. Descrete-29. Descrete-310. Descrete-411. Base States

Connection Form (Shak)Connection Form (Shak)

1. Principal Connection2. Compatible Connection3. Cartan Connection4. Ehesmann Connection5. Associated to Principal6. Global Connection Form7. Affine Connection

Principal Connection ( PURAJAB)Principal Connection ( PURAJAB)

• Let U be an open cover of M, along with G-frame on each U, denoted by eU.

• These one related on the intersection of overlapping open set by e V=ev.huu for some G-valued function h uu defined in Un V.

Compatible Connection (PABMAN)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 (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 ( CHITRAREK)

• Ehesmann connection is a version of the notion of a connection which is defined on ordinary fiber bundle.

Associated To Principle ( BAHURUP)Associated To Principle ( BAHURUP)

• A Principle G-connection w in a principal G-bundle PM gives rise to a collection of connection focus on M.

Global Connection Form ( BISHWAVAB)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)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)Curvature Form (KUSH)

1.Curvilinear Co-ordinate2.Non-tensor3.Parallel Transport4.Geodesics5.Co-variant Derivative6.Curvature Tensor7. Bianchi Identity

Curvilinear Co-ordinate (BASU)Curvilinear Co-ordinate (BASU)

• Curvilinear co-ordinates are co-ordinate in which angles between axes can changes from point to point.

Nontensors ( HARRUCHI)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)Parallel Transport ( NAVIGUPTA)

Geodesics (SUTABRATA)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)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 :=A1_AaTa

Curvature Tensor (BAMDEV)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)Bianchi Identity (BASUDAN)

• Following differential relation known as Bianchi identity

• R :+R :+R :=0

Torsion Form ( KOUNCH)Torsion Form ( KOUNCH)

1. Curvature Tensor2. Cyclic Sum3. Curvature Form4. Affinity Parameterized Geodesic5. Twisting of Reference Frame6. Torsion of a Filament7. Affine Developments

Curvature Tensor (AM)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)Cyclic Sum ( MADHUBRAHA)

(R(X,Y)X):=• R(X,Y)Z+R(Y,Z)X+R(Z,X)Y

Curvature Form (MEGHAPRISTA)Curvature Form (MEGHAPRISTA)

• Curvature form is the gl(n)-valued 2-form.

=Dw = dw +ww• D denotes the exterior covariant

derivative• 1. D=• 2. D=0

Affinely Parametrized Geodesic Affinely Parametrized Geodesic ( SUDHAMA)( 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)Twisting Of Reference Frame ( BHAJISTA)

• Frenet-Serret Formula describe how a particular moving frame twists along a curve.

Torsion Of A Filament ( LOHITAN)Torsion Of A Filament ( LOHITAN)

• Length –maximizing (geodesic) configuration and its energy-minimizing configuration.

Affine Developments (BANASPATY)Affine Developments (BANASPATY)

• Suppose that xt is a curve in M.The affine developments of xt is the unique curve ct in Tx0M such that

• Ċt=t0xt , c0=0

• Where t0:TxtMTx0M

• is the parallel transport associated to.

Solder Form (SHALMLI)Solder Form (SHALMLI)

1. Smooth Manifold2. Symplectic Form3. Complex Manifold4. Vierbein or Tetrad Theory5. Signature6. Palatini Action7. Interwiners

Smooth Manifold ( SURACHAN)Smooth Manifold ( SURACHAN)

• A differentiable manifold for which all the transitions maps are smooth.

• C

Symplectic Form ( SOUMANAS)Symplectic Form ( SOUMANAS)

• A symplectic form on a manifold M is a non-degenerative closed two form w.

Complex Manifold ( RAMANAK)Complex Manifold ( RAMANAK)

• Complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space.

Vierbeib Or Tetrad Theory ( DEVBARS)Vierbeib Or Tetrad Theory ( DEVBARS)

• Special case of application of Cretan connection in four – dimensional manifold.

Signature (PARIVADRA)Signature (PARIVADRA)

• Signature is the number of positive and negative Eigen values of the vector,

Palatini Action ( APAYAN)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 .

M

eeFS2

1

Interwiners (ABIGAN)Interwiners (ABIGAN)

• If X and Y are the representative spaces of two linear representations of G has a linear map f:XY 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)Levi-civita Connection (PLAKH)

1. Preserve the metric2. Derivative along curve3. Unit sphere in R34. Isomorphism5. Leibniz Rule6. Torsion Free7. Weitzenbock Connection

Preserve The Metric ( SHIV)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 (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 RUnit Sphere In R33 ( SHANT) ( 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)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)Leibniz RulE (AMRIT)

• C(S2) linear is first variable.• Consider a map f:S2R 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 ( 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)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)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 ( Derivative Of F Along The Vector Field X ( SWARNAPRASTHA)SWARNAPRASTHA)

• Given a function f:MR 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-Einstein Summation Convention 1-form( CHANDRASHEKHAR)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)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)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)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)Global Isometric ( SINGHAL)

• Bijective distance preserving maps.• Any reflection, translation, rotation is

a global isometry on Eucledean space.

Linear Isometric ( LANKA)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 (RAMNAK)

• Spin connection is a connection on a spin or bundle

bbaab eeeW

Descrete-1 (PUSKAR)Descrete-1 (PUSKAR)

1.CONVERGENCE SPACE2.UNIFORM SPACE

CONVERGENCE SPACE (GHATAK)CONVERGENCE SPACE (GHATAK)

UNIFORM SPACE (RAMNAK)UNIFORM SPACE (RAMNAK)

Descrete-2Descrete-2

1.Classica Module Space2.Quantum Module Space

CLASSICAL MODULLI SPACE (STHANU)CLASSICAL MODULLI SPACE (STHANU)

QUANTUM MODULLI SPACE(SARBANANDA)QUANTUM MODULLI SPACE(SARBANANDA)

Descrete-3Descrete-3

1.Product Space2.Compact Space3.Locally Compact Space

Descrete-4 (AGNIDH)Descrete-4 (AGNIDH)

1.Satadhenu2.Swayangbhoj3.Hyperbola Action4.Hyperbollic Space

(SATADHENU)(SATADHENU)

( SWAYANGBHOJ)( SWAYANGBHOJ)

Hyperbolic Motion ( KRITOBARMA)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)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)Base States (BHAJAMAN)

1. Path of spin-12. Path of spin-1A3. Path of spin-1B4. Path of spin-1C5. Filters in series6. Filters in series-A7. 3- Filters in series

Path Of Spin – 1 ( SATADHENU)Path Of Spin – 1 ( SATADHENU)

• Stern-Gerlach Apparatus

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Path Of Spin-1 A ( KRITABARMA)Path Of Spin-1 A ( KRITABARMA)

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Path Of Spin-1b ( KRITABARMA)Path Of Spin-1b ( KRITABARMA)

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Path Of Spin -1 C ( SWANGBHOJ)Path Of Spin -1 C ( SWANGBHOJ)

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FILTERS IN Series (SUR)FILTERS IN Series (SUR)

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Filters In Series-a (BIDURATH)Filters In Series-a (BIDURATH)

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3-filters In Series (SINI)3-filters In Series (SINI)

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M-theoryM-theory

• 11th dimension ( ROW & COLUMN).• Not a part of moving frames

M-theory (ROW)M-theory (ROW)

1.Type-i String2.Type-iia String3.Type-ii B String4.String With Loose Ends5. String With Loose Ends6. Nambu 3-aljebra( Lie 3-aljebra)7.Non-associative Generalisation Of Lie Aljebra8. Weakly Interacting9.Strongly Interacting10.Prosen11. Anu

Chiral (NIMLOCHI)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 NAMBU 3-ALJEBRA3-ALJEBRA (NIMLOCHI) (NIMLOCHI)

Non-associative Generalization Of Lie Non-associative Generalization Of Lie Algebra (BRISNI)Algebra (BRISNI)

• A binary operation on a set S does not satisfy the associative law is called non-associative.

• (XxY)xYXx(YxZ)• For any X,Y,Z S

Weak Interactions (KINGKIN)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)Strongly Interacting (SINI)

• Strong interaction holds quarks and gluons together to form protons and neutrons.

Type-I String (AYUJIT)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)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)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)String With Closed End (KAPATOROM)

• Strings with closed loops are free to move from membrane to membrane.

String With Loose Ends (BILOM)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 (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 (ANU)

• U-duality is a symmetry of sting theory or M-theory combining S-duality and T-duality transformation.

M-theory(COLUMN)M-theory(COLUMN)

1.T-duality2.S-duality3.Supergravity4.Dimensional Reduction To A Line5.Dimensional Reduction To A Circle6.Heteriotic String( E8xe8 + So(32)7.Heteriotic So(32)8.Joy9.Judhajit10.Juldhan

Anamoly (ANAMITRA)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 (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 (KANGBAL)

• Super gravity is a field theory that combines the principle of super symmetry and general relativity.

Anthropic Explanation (KONI)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 Dimensional Reduction To A Circle ( KONI)( KONI)

• Get Type IIA String theory

Holographic Principle ( IBPHALAK)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 Dimensional Reduction To A Line Segment ( EBOFALOK)Segment ( EBOFALOK)

• Get Heterotic SO(32) String Theory

TYPE IIA STRING (JUDHAJIT)TYPE IIA STRING (JUDHAJIT)

Heteriotic (E8XE8+ SO(32))(JUGDHAR)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) TYPE-IIA STRING (JUJUDHAN)

Heteriotic So (32) (KRUR) (AKRUR)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.