2. Differential Geometry and Curvature

Question 1

Topological Space

Definition

Answer 1

(X,Θ)

• have a set X, and Θ={Ui}, i∈I
• where Θ is a collection is special subsets of X called open sets, they obey the following rules:
  i) all unions of open sets are open
  ii) finite intersections of open sets are open
  iii) total set X and empty set Ø are open
• a different collection of subsets Ø may endow the same point space X with a different topology

Question 2

Spacetime

Definition

Answer 2

-a connected, Hausdorff, differentiable pseudo-Riemann manifold of dimension 4 whose points are called event

Question 3

Basis for a Topology

Definition

Answer 3

-a subset of all possible open sets which by intersections and unions can generate all possible open sets

Question 4

Open Cover

Definition

Answer 4

-an open cover {Ui} of X is a collection of open sets such that every point in x∈X is contained in at least one Ui

Question 5

Compact

Definition

Answer 5

-X is compact if every open cover has a finite sub cover

Question 6

Hausdorff

Definition

Answer 6

-X is Hausdorff if every pair of disjoint points is contained in a disjoint pair of open sets

Question 7

Neighbourhood

Definition

Answer 7

-any open set containing a point x∈X is also called a neighbourhood of x

Question 8

Continuous

Definition

Answer 8

-a function from one topological space X to another Y, f: X->Y, is continuous if the inverse image of every open set is open

Question 9

Homeomorphic

Definition

Answer 9

-two topological spaces are homeomorphic if there is a one-to-one map φ from X to Y (a bijection) such that both φ and φ^(-1) are continuous
-by Leibniz’s principle of the identity of indiscernibles,
two homeomorphic topological spaces are usually thought to be the same

Question 10

Smooth n-Dimensional Manifold

Definition

Answer 10

• we define a smooth n-dimensional manifold with a smooth atlas of charts as:
  i) a topological space X
  ii) an open cover of set {Ui} of X called patches
  iii) a set (atlas) of maps φi:Ui->ℝ^n called charts, which are injective, homeomorphisms onto their images and whose images are open in ℝ^n such that:
  iv) if two patches Ui and Uj intersect, then on Ui∩Uj, both ϕj◦ϕi^(-1) and ϕi◦ϕj^(-1) are smooth maps from ℝ^n to ℝ^n

Question 11

Local Coordinate

Definition

Answer 11

• we write ϕ(x) = xµ, with µ = 1,2,…,n

- xµ is called a local coordinate on X

Question 12

Compatible Atlas

Definition

Answer 12

-two atlases are said to be compatible if, where defined, the coordinates are smooth functions of each other

Question 13

Smooth n-Manifold with Complete Atlas

Answer 13

• a smooth n-manifold with complete atlas is the maximal equivalence class consisting of all possible compatible atlases
• denoted M or M^n

Question 14

Real Valued Smooth Function

Definition

Answer 14

-a function f, f : M −→ R, is a real valued sooth function if it is smooth in all coordinate systems; that is, if f◦ϕ^(-1) = f(xµ) is smooth

Question 15

C^∞(M)

Definition

Answer 15

• the set of all smooth functions on a manifold

- it forms a commutative ring

Question 16

Orientable

Definition

Answer 16

-a manifold M is said to be orientable if it admits an atlas such that for all overlaps the Jacobian satisfies:
det(∂xiµ/∂xjν) > 0

Question 17

Smooth Curve

Definition

Answer 17

• a smooth curve γ in M is a smooth map γ:ℝ->M

- in local coordinates, γ:s->xµ(s) where xµ is a smooth function of s

Question 18

Closed Curve

Definition

Answer 18

-a map from S to M

Question 19

Simple Curve

Definition

Answer 19

-a curve is simple if it is one-to-one onto its image

Question 20

Path

Definition

Answer 20

-a path is the image of a curve, that is, it is a point set
-if M is a spacetime, the path of a curve in M is called a world line, and corresponds
to a particle

Question 21

Curves vs Paths

Answer 21

• a curve contains information about the parameterisation

- a path does not

Question 22

Tangent Vector

Definition

Answer 22

-given a curve γ in M and a function f, compose them to get a map γ◦f:ℝ->ℝ
-given in local coordinates by, f(xµ(s))
-differentiate with respect to s
-if we look at this at a point p∈M and vary the curves passing through that point, we get a map T:C^∞(M)→ℝ
-where:
T: f->Tf = df/ds|s=0
-where xµ(0)
-T is called the tangent vector at p

Question 23

Tangent Vector

Properties

Answer 23

-a tangent vector T is a map satisfying:
i) linearity:
T(f+g) = T(f) + T(g)
ii) Leibniz’s rule:
T(fg) = T(f)g + fT(g)

Question 24

Tangent Space

Definition

Answer 24

-the space of tangent vectors at a point p∈M is a vector space, the tangent space denoted:
TpM or Tp(M)

Question 25

Vector Space

Definition

Answer 25

• a set V combined with a field F

- i.e. a set of elements in V which can be added an multiplied by scalars (numbers that belong to the field F)

Question 26

Tangent Space as a Vector Space

Answer 26

-a tangent space, TpM is a vector space of dimension n
-this can be shown by considering a Taylor expansion around x∈M, a point in the neighbourhood of p
-this tells us that ∂/∂xμ is a basis of the tangent space at p
-thus in local coordinates:
T = Tμ ∂/∂xμ
-if T is the tangent vector to a curve γ then:
Tμ = dxμ(s)/ds |s=0

Question 27

Vector Field

Definition

Answer 27

-a continuous assignment of a vector V(p)∈TpM to each point p in the manifold M
-can be written as:
V = Vμ(x) ∂/∂xμ
-the set of all vector fields on M is denoted by Γ(TM) or X(M)

Question 28

Integral Curves

Definition

Answer 28

-given a vector field V∈X(M), at least locally, the associated integral curves are defined as the solutions of the non-linear ordinary differential equations:
Vμ(x) = dxμ(s)/ds
-whose tangent vectors coincide with the vector field at every point in M

Question 29

Congruence of Curves

Definition

Answer 29

-in general, a family of curves passing through a given point p∈M is called a congruence of curves

Question 30

Tangent Bundle

Definition

Answer 30

-denoted, TM, the space of all possible vectors at all possible points:
TM = ⋃TpM
-the intersection of the tangent spaces for every point p∈M
-TM is a 2n-dimensional manifold with local coordinates (xμ,Vv) where V=Vv∂/∂xv
-a vector field can be thought of as a sort of n-dimensional surface in TM

Question 31

Dual Space Definition

Answer 31

• given any finite dimensional vector space V, we define its duals space V* as the space of linear maps V->ℝ
• we write for ω(u)=⟨ω|u⟩= ⟨ω,u⟩ ∈ℝ
• V* has the same dimension as V, elements of V* are called one-forms or co-vectors

Question 32

Basis for Dual Space

Answer 32

-given a basis {eμ} for V, we define ωμ=ω(eμ)
-in this way, if a vector v∈V has components vμ in the basis {eμ}, then we see that:
ω(v) = ωμ vμ
-we define the dual space basis {eμ} such that:
⟨eμ|ev⟩ = δvμ

Question 33

One-Forms

Answer 33

• geometrically, vectors define directions through the origin of V
• can think of one-forms as hyperplanes or co-normal planes through the origin

Question 34

Cotangent Space

Definition

Answer 34

-at every point p∈M one defines the cotangent space Tp*M as the dual space of the tangent space TpM
-then a one-form (or co-vector) is one for which:
ω(fU) = fω(U)
-with f a function in M and U∈Χ(M)

Question 35

Space of One Forms

Answer 35

-we call Ω^1(M) the space of one-forms on a manifold M
-and Ω^0(M) the set of real-valued functions on M, the same set as C^∞(M)
-then we have a map:
d : C^∞(M)=Ω^0(M) -> Ω^1(M)
-such that f->df