, 
open. 
What is an affine connection?
An affine connection is a connection 
satisfying
i)
ii)
iii)
satisfying
i)
ii)
iii)
What is a Riemannian or Levi Civita connection?
An affine connection that also satisfies
torsion-free
torsion-free
if connection is torsion free.
have?
Jacobi Identity
product rule
?
symmetries
Bianchi identityHow is a differentiable manifold of dimension n defined?
A set M, a family of injective maps 
is open such that
are open and 
is differentiable 
.
there exists a maximal atlas {
} relative to M
is open such that are open and is differentiable . there exists a maximal atlas {} relative to MHow is a Riemannian Metric defined?
As g or 
as an inner product,
where an inner product is
-symmetric,
-linear and
-positive definite
as an inner product,where an inner product is
-symmetric,
-linear and
-positive definite
a local isometry?
is an Isometry if
Let 
be a Vector field for 
. When is X differentiable and how is 
for 
defined?
be a Vector field for . When is X differentiable and how is for defined?X is differentiable if 
is differentiable.
is differentiable.How was the covariant Derivative for two vector fields 
simply (firstly) calculated?
simply (firstly) calculated?
where
be calculated in terms of 
?
defined?
How is a topological space 
defined?
defined?We have a Family 
of open subsets of 
satisfying
(T.1)
(T.2)
(T.3)
of finite i's
of open subsets of satisfying(T.1)
(T.2)
(T.3)
of finite i'sWhat is the algebraic Definition of a tangent vector at p?
satisfying(1)
(2)
where
{diff. fcts. at p}An n-dimensional smooth manifold M is called orientable if
There exists a differentiable atlas 
for M s.t.
at p, for all p in 
for M s.t. at p, for all p in 
be connected. What do we know for M?What is an Immersion?
where 
is differentiable is called an immersion if 
is injective 
equivalently
What is an Embedding?
An Immersion 
that is a homeomorphism 
cont's)
and
has the subspace topology induced from 
.
that is a homeomorphism cont's) and
has the subspace topology induced from .
defined?What is an intrinsic quality?
A geometric quality that can be expressed only in terms of 
and its derivative function.
and its derivative function.What is the difference of an affine connection and the covariant derivative?
is an affine connection as 
is the covariant derivative for the vector fields X and Y.
calculated?
defined?
is called a tangent vector to 
at 
if 
with
where
?
where
is any Vector.
, 
.
,
open, 
smooth and
or 
are the Eigenvalues of 
.
Kartensatzinfo:
Autor: Yann-Paul
Oberthema: Mathematik
Thema: Differentialgeometrie123
Veröffentlicht: 15.11.2018
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