2. Differential Geometry and Curvature Flashcards Preview

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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

Spacetime
Definition

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

3

Basis for a Topology
Definition

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

4

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

Compact
Definition

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

6

Hausdorff
Definition

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

7

Neighbourhood
Definition

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

8

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

9

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

10

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

11

Local Coordinate
Definition

-we write ϕ(x) = xµ, with µ = 1,2,...,n
-xµ is called a local coordinate on X

12

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

14

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

16

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

17

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

18

Closed Curve
Definition

-a map from S to M

19

Simple Curve
Definition

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

20

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

21

Curves vs Paths

-a curve contains information about the parameterisation
-a path does not

22

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)

24

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)

25

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)

26

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

27

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

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

29

Congruence of Curves
Definition

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

30

Tangent Bundle
Definition

-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