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Differentialgeometrie123 (40 Cards)

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What is a level set?

,
where

open.

What is the Hessian operator

What is an affine connection?

An affine connection is a connection

satisfying
i)

ii)

iii)

What is a Riemannian or Levi Civita connection?

An affine connection that also satisfies

torsion-free

How can the Torsion be calculated?

if connection is torsion free.

Which properties does the Lie bracket

have?
4 Properties

Jacobi Identity

product rule

What are the symmetries and the Bianchi Identity of

?

symmetries

Bianchi identity

How 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

How is a Riemannian Metric defined?

As g or

as an inner product,
where an inner product is
-symmetric,
-linear and
-positive definite

How does the

as a Riemannian metric look like?

When is

a local isometry?

If

How is an Isometry for a Riamannian metric defined?

is an Isometry if

Let

be a Vector field for

. When is X differentiable and how is

for

defined?

X is differentiable if

is differentiable.

How was the covariant Derivative for two vector fields

simply (firstly) calculated?

where

How were the Christoffel Symbols defined?

where

How can the Christoffelsymbols

be calculated in terms of

and

?

How is the Riemann tensor

calculated?

How is the Riemannian tensor for vector fields

defined?

How is a topological space

defined?

We have a Family

of open subsets of

satisfying
(T.1)

(T.2)

(T.3)

of finite i's

What 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

Let

be connected. What do we know for M?

M is orientable

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

.

How is the Lie bracket

defined?

where

is a vector field.

What is an intrinsic quality?

A geometric quality that can be expressed only in terms of

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.

How is

calculated?

How is a tangent vector to

defined?

is called a tangent vector to

at

if

with

How to calculate standard unit normal field N?

where

What are the properties of

?

What is an Integral curve?

A diff. b. curve

where

where

is any Vector.

What is a level set SURFACE

,
where

.

What is a parametrized space in

,

where

open,

smooth and

or

How to calculate the Weingarten map for smooth

How to calculate normal curvature of S at p in the direction

?

How to calculate normal curvature for curves?

How to calculate Mean curvature for

How to calculate Gauß-curvature for

?

What are the principial curvatures?

where

are the Eigenvalues of

.

Flashcard set info:

Author: Yann-Paul

Main topic: Mathematik

Topic: Differentialgeometrie123

Published: 15.11.2018

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