Flashcards in 2. Differential Geometry and Curvature Deck (35)

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1

##
Topological Space

Definition

###
(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

2

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Spacetime

Definition

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

3

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Basis for a Topology

Definition

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

4

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Open Cover

Definition

### -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

5

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Compact

Definition

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

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Hausdorff

Definition

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

7

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Neighbourhood

Definition

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

8

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Continuous

Definition

### -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

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Homeomorphic

Definition

###
-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

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Smooth n-Dimensional Manifold

Definition

###
-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

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Local Coordinate

Definition

###
-we write ϕ(x) = xµ, with µ = 1,2,...,n

-xµ is called a local coordinate on X

12

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Compatible Atlas

Definition

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

13

## Smooth n-Manifold with Complete Atlas

###
-a smooth n-manifold with complete atlas is the maximal equivalence class consisting of all possible compatible atlases

-denoted M or M^n

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Real Valued Smooth Function

Definition

### -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

15

##
C^∞(M)

Definition

###
-the set of all smooth functions on a manifold

-it forms a commutative ring

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Orientable

Definition

###
-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

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Smooth Curve

Definition

###
-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

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Closed Curve

Definition

### -a map from S to M

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Simple Curve

Definition

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

20

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Path

Definition

###
-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

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## Curves vs Paths

###
-a curve contains information about the parameterisation

-a path does not

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Tangent Vector

Definition

###
-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

23

##
Tangent Vector

Properties

###
-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)

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Tangent Space

Definition

###
-the space of tangent vectors at a point p∈M is a vector space, the tangent space denoted:

TpM or Tp(M)

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Vector Space

Definition

###
-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)

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## Tangent Space as a Vector Space

###
-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

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Vector Field

Definition

###
-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)

28

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Integral Curves

Definition

###
-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

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Congruence of Curves

Definition

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

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