Search results
ment of the euclidean geometry is clearly shown; for example, it is shown that the whole of the euclidean geometry may be developed without the use of the axiom of continuity; the signifi-cance of Desargues’s theorem, as a condition that a given plane geometry may be regarded as a part of a geometry of space, is made apparent, etc. 5.
Lecture 3: Geometry Geometry is the science of shape, size and symmetry. While arithmetic deals with numerical structures, geometry handles metric structures. Geometry is one of the oldest mathematical disci-plines. Early geometry has relations with arithmetic: the multiplication of two numbers n mas an
- 1.1 De nition of manifolds
- What is geometry?
- 1.5 The theorem of Sard
- y V 2
- (x + z)g(z) dz
- ! = ! dx ;
- 2.7 Theorem of Stokes for chains
- d LX (x)
- T(X; Y ) = rXY rY X [X; Y ] = 0 :
- Xg(Y; Z) = g(rXY; Z) + g(Y; rXZ)
- Theorem 3.6.1 (Fundamental theorem of Riemannian geometry)
- Xg(Y; Z) Y g(Z; X) Zg(X; Y )
- Rijkl = K(Xi; Xj; Xk; Xl) :
- Lemma 3.11.3 (The Jacobi identity)
- L[X;Y ] = LXLY LY LX
- R(X; Y )Z R(Y; Z)X R(Z; X)Y
- Y g(Z; rXW) + g(Z; rY rXW) ;
- 3.12 Ricci tensor and scalar curvature
- R(Xk; Xm)Xn = Xj Rj kmnXj = Xj Rj nkm :
- Ui = Ri kljAj xk xl :
- ✪ Aj xk xl xm : @xm
- ✪ mkj + + = 0 : @xm @xk @xl
- i = Xi Bk fi
- 2) (*) Topic: Tensors.
- Bk
- l fij :
- ( 1) (e (1); : : : ; e (n 1))An
- ZG f dV = Z f ;
- + B3(x)dx2 ^ dx3 ^ B2(x)dx2 ^ dx4 + B1(x)dx3 ^ dx4
- Einstein tensor
- = 1=2R j
- Week 8 (Solutions)
- + B3(x)dx2 ^ dx3 B2(x)dx2 ^ dx4 + B1(x)dx3 ^ dx4 :
- Appendices
- v w = v(w) = viwjei(ej) = Xj vjwj :
- (Av)j = Xi Aj vi
- i ej ;
De nition. A locally Euclidean space M of dimension n is a Hausdor topological space, for which each point x M has a neighborhood U, which is homeomorphic to an open subset (U) of Rn. The pair 2 (U; ) is called a coordinate system or a chart. De nition. A Ck atlas on a locally Euclidean space M is a collection
The partition of mathematics into topics is a matter of fashion and depends on the time period. It is therefore not so easy to de ne what part of mathematics is geometry. The original meaning of geometry origins in the pre-Greek antiquity, where measure-ment of the earth had priority. However, in ancient Greek, most mathematics was considered geome...
De nition. A subset A of Rn has measure zero, , if there exists for every > 0 a countable open cover Ui of A such that the sum of the Euclidean volumes of Ui is less than . A subset A of a nite-dimensional manifold M is of measure zero, if for each chart : U ! Rn the set (U \ A) has measure zero. Remark. Because the coordinate changes ij between tw...
x and ✪ has a compact support which is contained in a set V U
which is obviously smooth because Dkg(x) = (x + compact support, there exists a constant C such that RRn(Dk We get jjDg(x)ujj z))g(z) dz. Because g has
ZA (!)(x) = det(D (x))!( (x)). De nition. Let be a p-form on M and a p-simplex. We can pull back the form to a neighborhood of p in Rn. De ne the integration of the p-form on a p-simplex as = Z :
The following theorem is a generalization of the fundamental theorem of calculus, which says f = for a function on a one dimensional manifold. It also generalizes the theorem of Green in the plane or the theorems of Stokes or Gauss in the three dimensional R ✪
= ( (x) (x))=t : dt It is called De nition. Let as ✪ the Lie derivative with respect to the vector eld X.
Example. The canonical connection on Euclidean space is rXY = DY (X) = Xj@jY i. De nition. A connection on a pseudo Riemannian manifold (M; g) is called Pj compatible with the metric if for all vector elds X; Y; Z, the identity ✪
holds. (This de nition is motivated by the Leibniz rule.) ✪ ✪
On every pseudo Riemannian manifold (M; g), there exists a unique connection r which is compatible with the metric. ✪ Proof. Because r is compatible with the metric, we obtain the three identities ✪ Adding the
= g(rXY; Z) + g(Y; rXZ) ; = g(rY Z; X) + g(Z; rY X) ; = g(rZX; Y ) + g(X; rXY ) : rst two equalities and subtracting the third one and using that free, this gives is torsion ✪ 2g(rXY; Z) = Xg(Y; Z) + Y g(Z; X) Zg(X; Y ) + g([X; Y ]; Z) g([X; Z]; Y ) g([Y; Z]; X) ✪ which shows uniqueness. The right hand side is linear and is for xed X; Y a tensor in...
(It is custom to take here the letter In the R and not K). following theorem, we need the identity ✪
[X; [Y; Z]] + [Y; [Z; X]] + [Z; [X; Y ]] : Proof. By de nition, we have
so that L[X;[Y;Z]] = LXL[Y;Z] L[Y;Z]LX = LXLY LZ LXLZLY LY LZLX + LZLY LX L[Y;[Z;X]] = LY L[Z;X] L[Z;X]LY = LY LZLX LY LXLZ LZLXLY + LXLZLY L[Z;[X;Y ]]
gives on the right hand side = = = rXrY Z rY rXZ r[X;Y ]Z rY rZX rZrY X r[Y;Z]X rZrXY rXrZY r[Z;X]Y rX(rY Z rZY )+rY (rZX rXZ)+rZ(rXY rY X) This is, because the connection is torsion free, r[X;Y ]Z r[Y;Z]X r[Z;X]Y : rX[Y; Z] + rY [Z; X] + rZ[X; Y ] r[X;Y ]Z and again because the connection is torsion free r[Y;Z]X r[Z;X]Y [X; [Y; Z]] + [Y; [Z; X]] +...
We see that g(rY rXZ; W) g(r[X;Y ]Z; W) = = Y g(rXZ; W) g(rXZ; rY W) = Y Xg(Z; W) Y g(Z; rXW) Xg(Z; rY W) + g(Z; rXrY W) ; (XY Y X)g(Z; W) g(Z; r[X;Y ]W) : g(R(X; Y )Z; W) = g(Z; R(X; Y )W) which establishes the second identity. c) Take Xi so that Rijkl = K(Xi; Xj; Xk; Xl). Rijkl = Rjkil Rkijl (a) = Rjkli + Rkilj (b) = Rklji Rljki Rilkj Rlkij (a) =...
Let us rst compute the Riemann curvature tensor in coordinates using Xi = = xi. De ne Rk;m;n;j = R(Xk; Xm; Xn; Xj), so that ✪
(The second expression is often used also and is the same like the middle expression because of the symmetry of R). ✪ ✪
The sum of the parallel transports around this plaquette Pkl(x) and the parallel plaquette Pkl(x + xm) is @Ri klj Ui =
If we sum up all parallel transports around the cube, then each parallel transport around an edge has been added twice with opposite sign. That is
Because we have used normal coordinates, this is the Bianchi identity.
: b) fk ~ = f(~ek) = f(Aek) = f(Xi Ai kei) = Xi Ai keif(ei) = Xi Ai kfi :
a) fkl ~ = f(~ ek; el) ~ = f(Bek; Bel) = f(Xi Bk ei; i Xj Bl jej) = BkBl i jf(ei; ej) =
i Bl jfij : b) fkl ~ = f(~ ek; el) ~ = f(Aek; Ael) = f(Xi Ai kei; Xj Aj ej) = l Ai kAj
c) fk ~ = f(~ek; ~el) = f(Bek; Ael) = f(Xi Bk i ei; Xj l Aj ej) = l Bk i Aj l f(ei; ej) =
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
Adjoint map because for each ek, the left hand side is ek Aei = Ak , while the right hand side is ek i Pn ✪
- 661KB
- 121
mathematics courses as real analysis and abstract algebra. A course based on this book will enrich the education of all mathematics majors and will ease their transition into more advanced mathematics courses. The book also includes emphases that make it especially appropriate as the textbook for a geometry course taken by future high school
- 1MB
- 121
What Is Geometry? Geometry is the visual study of shapes, sizes, patterns, and positions. It occurred in all cultures, through at least one of these five strands of human activities: 1. building/structures (building/repairing a house, laying out a garden, making a kite, …) 2.
- 19KB
- 2
Chapter 1. Geometry and Space Section 1.1: Space, distance, geometrical objects A point P on the real line is labeled by a single coordinate P = x, a point in the plane is fixed by two coordinates P = (x,y) and a point in space is determined by three coordinates P = (x,y,z). Depending on which coordinates are positive, one can divide the line ...
People also ask
Where did geometry come from?
Is geometry a part of mathematics?
What are the elements of geometry called?
Why should you read a book about geometry?
Why do we need a geometry course?
What chapters should be included in a geometry course?
A vector is an oriented segment. More precisely, a vector is an equivalence class of oriented segments, all parallel, congruent, and pointing in the same direction. A vector is denoted by a lower case letter with an arrow on top of it, or by two upper case letters, the endpoints, with an arrow on top. Definition.
