2. Differential Geometry and Curvature Flashcards
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
Spacetime
Definition
-a connected, Hausdorff, differentiable pseudo-Riemann manifold of dimension 4 whose points are called event
Basis for a Topology
Definition
-a subset of all possible open sets which by intersections and unions can generate all possible open sets
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
Compact
Definition
-X is compact if every open cover has a finite sub cover
Hausdorff
Definition
-X is Hausdorff if every pair of disjoint points is contained in a disjoint pair of open sets
Neighbourhood
Definition
-any open set containing a point x∈X is also called a neighbourhood of x
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
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
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
Local Coordinate
Definition
- we write ϕ(x) = xµ, with µ = 1,2,…,n
- xµ is called a local coordinate on X
Compatible Atlas
Definition
-two atlases are said to be compatible if, where defined, the coordinates are smooth functions of each other
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
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
C^∞(M)
Definition
- the set of all smooth functions on a manifold
- it forms a commutative ring
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
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
Closed Curve
Definition
-a map from S to M
Simple Curve
Definition
-a curve is simple if it is one-to-one onto its image
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
Curves vs Paths
- a curve contains information about the parameterisation
- a path does not
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
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)
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)
