aarjaneiro/planetmath huggingface dataset:
tilt curve

tilt curve

The

tilt curves

(in German

die Neigungskurven

) of a surface

z

=

f

⁢

(

x

,

y

)

are the curves on the surface which intersect (

http://planetmath.org/ConvexAngle

) orthogonally the

level curves

f

⁢

(

x

,

y

)

=

c

of the surface.  If the gravitation acts in direction of the

negative

z

-axis, then a drop of water on the surface aspires to slide along a tilt curve.  For example, since the level curves of the sphere

z

=

±

r

2

-

x

2

-

y

2

are the “

latitude

circles”, the tilt curves of the sphere are the “

meridian

circles”.  The tilt curves of a

helicoid

are

circular helices

.

If the tilt curves are projected on the

x

⁢

y

-plane, the

differential equation

of those projection curves is

d

⁢

y

d

⁢

x

=

f

y

′

⁢

(

x

,

y

)

f

x

′

⁢

(

x

,

y

)

.

(1)

Naturally, they also cut orthogonally (the projections of) the level curves.

Example.

Let us find the tilt curves of the

elliptic paraboloid

z

=

x

2

a

2

+

y

2

b

2

.

The level curves are the

ellipses

x

2

a

2

+

y

2

b

2

=

c

.  Now we have

f

x

′

⁢

(

x

,

y

)

=

∂

∂

⁡

x

⁢

(

x

2

a

2

+

y

2

b

2

)

=

2

⁢

x

a

2

,

f

y

′

⁢

(

x

,

y

)

=

∂

∂

⁡

y

⁢

(

x

2

a

2

+

y

2

b

2

)

=

2

⁢

y

b

2

,

whence the differential equation of the tilt curves is

d

⁢

y

d

⁢

x

=

a

2

b

2

⋅

y

x

.

The

separation of variables

and the integration yield

∫

d

⁢

y

y

=

a

2

b

2

⁢

∫

d

⁢

x

x

,

then

ln

⁡

|

y

|

=

a

2

b

2

⁢

ln

⁡

|

x

|

+

ln

⁡

|

C

|

=

ln

⁡

(

|

C

|

⁢

|

x

|

a

2

/

b

2

)

,

and finally

y

=

C

⁢

|

x

|

a

2

/

b

2

.

(2)

Here, we may allow for

C

all positive and negative values.  The curves (2) originate from the origin and continue infinitely far.

Remark.

Given an arbitrary family of

parametre curves

on a surface

r

→

=

(

x

⁢

(

u

,

v

)

,

y

⁢

(

u

,

v

)

,

z

⁢

(

u

,

v

)

)

⊺

of

ℝ

3

, e.g. in the form

d

⁢

u

d

⁢

v

=

f

⁢

(

u

,

v

)

,

the family of its

orthogonal curves

on the surface has in the Gaussian coordinates

u

,

v

the differential equation

d

⁢

v

d

⁢

u

=

-

g

11

+

g

12

⁢

f

⁢

(

u

,

v

)

g

12

+

g

22

⁢

f

⁢

(

u

,

v

)

,

(3)

where

g

11

=

r

→

u

′

⋅

r

→

u

′

,

g

12

=

r

→

u

′

⋅

r

→

v

′

,

g

22

=

r

→

v

′

⋅

r

→

v

′

are the fundamental quantities

E

,

F

,

G

of Gauss, respectively.

Title

tilt curve

Canonical

name

TiltCurve

Date of creation

2013-03-22 18:08:22

Last modified on

2013-03-22 18:08:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A05

Classification

msc 53A04

Classification

msc 51M04

Related topic

OrthogonalCurves

Related topic

Gradient

Related topic

PositionVector

Related topic

FirstFundamentalForm

Related topic

LineOfCurvature

Generated on Fri Feb 9 22:24:03 2018 by

LaTeXML
isocline

isocline

Let

Γ

be a family of plane curves.  The

isocline

of

Γ

is the locus of the points, in which all members of

Γ

have an equal slope.

If the family

Γ

has the

differential equation

F

⁢

(

x

,

y

,

d

⁢

y

d

⁢

x

)

=

0

,

then the equation of any isocline of

Γ

has the form

F

⁢

(

x

,

y

,

K

)

=

0

where

K

is .

For example, the family

y

=

e

C

⁢

x

of

exponential

(

http://planetmath.org/ExponentialFunction

) curves satisfies the differential equation

d

⁢

y

d

⁢

x

=

C

⁢

e

C

⁢

x

or

d

⁢

y

d

⁢

x

=

C

⁢

y

,  whence the isoclines are

C

⁢

y

=

K

,  i.e. they are horizontal lines.

http://en.wikibooks.org/wiki/Differential_Equations/Isoclines_1

Wiki

Title

isocline

Canonical

name

Isocline

Date of creation

2013-03-22 18:05:52

Last modified on

2013-03-22 18:05:52

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A25

Classification

msc 53A04

Classification

msc 51N05

Related topic

OrthogonalCurves

Generated on Fri Feb 9 22:24:05 2018 by

LaTeXML
differential logic

differential logic

Differential logic

is the component of logic whose object is the description of variation — for example, the aspects of change,

difference

,

distribution

(

http://

planetmath

.org/Distribution

), and diversity — in

universes of discourse

that are subject to logical description. In

formal logic

, differential logic treats the principles that govern the use of a

differential

logical

calculus

, that is, a

formal system

with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple example of a differential logical calculus is furnished by a

differential propositional calculus

(

http://planetmath.org/DifferentialPropositionalCalculus

). This extends an ordinary

propositional calculus

(

http://planetmath.org/PropositionalCalculus

) in the same way that

differential calculus

(

http://planetmath.org/Calculus

) extends

analytic geometry

.

1

Readings

•

Awbrey, J., “

http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction

Differential Logic :

Introduction

”.

•

Awbrey, J., “

http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0

Differential Logic and Dynamic Systems”.

Title

differential logic

Canonical

name

DifferentialLogic

Date of creation

2013-08-22 21:53:01

Last modified on

2013-08-22 21:53:01

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

16

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 53A40

Classification

msc 39A12

Classification

msc 34G99

Classification

msc 03B44

Classification

msc 03B42

Classification

msc 03B15

Related topic

DifferentialPropositionalCalculus

Related topic

DifferentialPropositionalCalculusExamples

Related topic

DifferentialPropositionalCalculusAppendices

Related topic

DifferentialPropositionalCalculusAppendix2

Related topic

DifferentialPropositionalCalculusAppendix3

Related topic

DifferentialPropositionalCalculusAppendix4

Generated on Fri Feb 9 22:24:23 2018 by

LaTeXML
axial vector

axial vector

Axial vector

behaves like a vector (

polar vector

) except that it is

invariant

under the

inversion

of its

coordinate

axes. For example, the

cross product

of two vectors

𝔸

and

𝔹

,

𝔸

×

𝔹

is an axial vector.
The cross product of two axial vectors stays as an axial vector, whereas the cross product of an axial vector and a polar vector turns to be a polar vector.

Title

axial vector

Canonical name

AxialVector

Date of creation

2013-03-22 15:15:00

Last modified on

2013-03-22 15:15:00

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 53A45

Related topic

vector

Related topic

tensor

Related topic

scalar

Related topic

Vector

Defines

pseudovector

Generated on Fri Feb 9 22:25:13 2018 by

LaTeXML
closed (differential form)

closed (differential form)

A

differential form

η

is called

closed

if its

exterior derivative

d

⁢

η

=

0

.

Title

closed (differential form)

Canonical

name

CloseddifferentialForm

Date of creation

2013-03-22 13:12:21

Last modified on

2013-03-22 13:12:21

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 53A45

Synonym

closed form

Synonym

closed differential form

Generated on Fri Feb 9 22:25:23 2018 by

LaTeXML
difference of vectors

difference of vectors

Let

a

→

and

b

→

be two vectors in the plane (or in a

vector space

).  The

difference vector

or

difference

a

→

-

b

→

of

a

→

and

b

→

is a vector

d

→

such that

b

→

+

d

→

=

a

→

.

Thus we have

b

→

+

(

a

→

-

b

→

)

=

a

→

.

(1)

According to the procedure of forming the

sum of vectors

by setting the addends one after the other, the equation (1) tallies with the picture below; when the minuend and the subtrahend emanate from a common initial point, their difference vector can be directed from the terminal point of the subtrahend to the terminal point of the minuend.

Remark.

It is easily seen that the difference

a

→

-

b

→

is same as the sum vector

a

→

+

(

-

b

→

)

where

-

b

→

is the

opposite vector

of

b

→

:  it may be represented by the directed line segment from the terminal point of

b

→

to the initial point of

b

→

.

Title

difference of vectors

Canonical

name

DifferenceOfVectors

Date of creation

2013-03-22 17:47:19

Last modified on

2013-03-22 17:47:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A45

Synonym

vector difference

Synonym

vector subtraction

Related topic

CommonPointOfTriangleMedians

Related topic

Difference2

Related topic

ProvingThalesTheoremWithVectors

Related topic

VectorValuedFunction2

Defines

difference vector

Defines

opposite vector

Generated on Fri Feb 9 22:26:05 2018 by

LaTeXML
Euclidean distance

Euclidean distance

If

u

=

(

x

1

,

y

1

)

and

v

=

(

x

2

,

y

2

)

are two points on the plane, their

Euclidean distance

is given by

(

x

1

-

x

2

)

2

+

(

y

1

-

y

2

)

2

.

(1)

Geometrically, it’s the length of the

segment

joining

u

and

v

, and also the norm of the difference vector (considering

ℝ

n

as

vector space

).

This

distance

induces a metric (and therefore a

topology

) on

ℝ

2

, called

Euclidean metric (on

R

2

)

or

standard metric (on

R

2

)

. The topology so

induced

is called

standard topology

or

usual topology on

R

2

and one basis can be obtained considering the set of all the open balls.

If

a

=

(

x

1

,

x

2

,

…

,

x

n

)

and

b

=

(

y

1

,

y

2

,

…

,

y

n

)

, then formula

1

can be generalized to

ℝ

n

by defining the Euclidean distance from

a

to

b

as

d

⁢

(

a

,

b

)

=

(

x

1

-

y

1

)

2

+

(

x

2

-

y

2

)

2

+

⋯

+

(

x

n

-

y

n

)

2

.

(2)

Notice that this distance coincides with

absolute value

when

n

=

1

.
Euclidean distance on

ℝ

n

is also a metric (

Euclidean

or standard metric), and therefore we can give

ℝ

n

a topology, which is called the standard (canonical, usual, etc) topology of

ℝ

n

. The resulting (topological and vectorial) space is known as

Euclidean space

.

This can also be done for

ℂ

n

since as set

ℂ

=

ℝ

2

and thus the metric on

ℂ

is the same given to

ℝ

2

, and in general,

ℂ

n

gets the same metric as

R

2

⁢

n

.

Title

Euclidean distance

Canonical name

EuclideanDistance

Date of creation

2013-03-22 12:08:21

Last modified on

2013-03-22 12:08:21

Owner

drini (3)

Last modified by

drini (3)

Numerical id

15

Author

drini (3)

Entry type

Definition

Classification

msc 53A99

Classification

msc 54E35

Synonym

Euclidean metric

Synonym

standard metric

Synonym

standard topology

Synonym

Euclidean

Synonym

canonical topology

Synonym

usual topology

Related topic

Topology

Related topic

BoundedInterval

Related topic

EuclideanVectorSpace

Related topic

DistanceOfNonParallelLines

Related topic

EuclideanVectorSpace2

Related topic

Hyperbola2

Related topic

CassiniOval

Generated on Fri Feb 9 22:26:36 2018 by

LaTeXML
affine connection

affine connection

An

affine connection

is a

connection

defined on the

tangent bundle

of a

manifold

. Such connections enjoy many

properties

which are not true of connections on other bundles. The reason for this goes back to the fact that to define a connection, one needs to consider two vector spaces, the

tangent space

and the

fiber space

. In the case of an affine connection, these two spaces are the same, and this leads to possibilities which one would not have when the spaces were distinct.

Perhaps the most important property of affine connections is that they allow one to define

geodesics

. Because the defining equation of a geodesic involves the

covariant derivative

of a tangent vector field, only affine connections give rise to geodesics. Furthermore, the converse is true: An affine connection is uniquely determined by the set of its geodesic curves.

Given an affine connection, one can obtain another affine connection by interchanging the role of the vector which specifies direction of

parallel transport

and the vector being transported. In local coordinates, an

expression

for this new connection is

C

′

j

⁢

k

i

=

C

k

⁢

j

i

Since the

difference

of two connections is a

tensor field

, one has a tensor field, known as the

torsion tensor

, associated to every affine connection.

T

j

⁢

k

i

=

C

j

⁢

k

i

-

C

k

⁢

j

i

Title

affine connection

Canonical

name

AffineConnection

Date of creation

2013-03-22 14:36:53

Last modified on

2013-03-22 14:36:53

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B05

Generated on Fri Feb 9 22:26:41 2018 by

LaTeXML
conjugate points

conjugate points

Let

M

be a

manifold

on which a notion of

geodesic

is defined. (For instance,

M

could be a

Riemannian manifold

,

M

could be a manifold with

affine connection

, or

M

could be a Finsler space.)

Two distinct points,

P

and

Q

of

M

are said to be

conjugate points

if there exist two or more distinct geodesic segments having

P

and

Q

as

endpoints

.

A simple example of conjugate points are the north and

south poles

of a sphere (endowed with the usual metric of constant curvature) — every

meridian

is a geodesic segment having the poles as endpoints.

Title

conjugate points

Canonical name

ConjugatePoints

Date of creation

2013-03-22 14:35:40

Last modified on

2013-03-22 14:35:40

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B05

Generated on Fri Feb 9 22:26:43 2018 by

LaTeXML
Levi-Civita connection

Levi-Civita connection

On any

Riemannian manifold

⟨

M

,

g

⟩

, there is a unique torsion-free

affine connection

∇

on the

tangent bundle

of

M

such that the

covariant derivative

of the

metric tensor

g

is zero, i.e.

g

is covariantly constant. This condition can be also be expressed in terms of the inner product operation

⟨

,

⟩

:

T

M

×

T

M

→

ℝ

induced

by

g

as follows: For all
vector fields

X

,

Y

,

Z

∈

T

⁢

M

, one has

X

⁢

(

⟨

Y

,

Z

⟩

)

=

⟨

∇

X

⁡

Y

,

Z

⟩

+

⟨

Y

,

∇

X

⁡

Z

⟩

and

∇

X

⁡

Y

-

∇

Y

⁡

X

=

[

X

,

Y

]

This

connection

is called the

Levi-Civita connection

.

In

local coordinates

{

x

1

,

…

,

x

n

}

, the

Christoffel symbols

(

http://planetmath.org/Connection

)

Γ

j

⁢

k

i

are determined by

g

i

⁢

ℓ

⁢

Γ

j

⁢

k

i

=

1

2

⁢

(

∂

⁡

g

j

⁢

ℓ

∂

⁡

x

k

+

∂

⁡

g

k

⁢

ℓ

∂

⁡

x

j

-

∂

⁡

g

j

⁢

k

∂

⁡

x

ℓ

)

.

Title

Levi-Civita connection

Canonical name

LeviCivitaConnection

Date of creation

2013-03-22 13:59:25

Last modified on

2013-03-22 13:59:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B05

Generated on Fri Feb 9 22:26:54 2018 by

LaTeXML
Riemann normal coordinates

Riemann normal coordinates

Riemann normal coordinates

may be thought of as a generalization of

Cartesian coordinates

from

Euclidean space

to any

manifold

(which should be at least twice

differentiable

) with

affine connection

. (Including

Riemannian manifolds

as a special case, of course!)

To define a system of Riemann normal coordinates, one needs to pick a point

P

on the manifold which will serve as origin and a basis for the

tangent space

at

P

. Suppose that the manifold is

d

dimensional. To any

d

-tuplet of

real numbers

(

x

1

,

…

⁢

x

n

)

, we shall assign a point

Q

of the manifold by the following procedure:

Let

v

be the vector whose components with respect to the basis chosen for the tangent space at

P

are

x

1

,

…

⁢

x

n

. There exists a unique

affinely-parameterized

geodesic

C

⁢

(

t

)

such that

C

⁢

(

0

)

=

P

and

[

d

⁢

C

⁢

(

t

)

/

d

⁢

t

]

t

=

0

=

v

. Set

Q

=

C

⁢

(

1

)

. Then

Q

is defined to be the point whose Riemann normal coordinates are

(

x

1

,

…

⁢

x

n

)

.

Riemann normal coordinates enjoy several important properties:

1.

The

connection coefficients

vanish at the origin of Riemann normal coordinates.

2.

Covariant derivatives

reduce to

partial derivatives

at the origin of Riemann normal coordinates.

3.

The partial derivatives of the components of the

connection

evaluated at the origin of Riemann normal coordinates equals the components of the curvature tensor. In fact, some authors take this property as a definition of the curvature tensor.

To every point on the manifold one may associate an open neighborhood of that point in which Riemann normal coordinates based at the point provide a

diffeomorphism

between the

neighborhood

and a subset of

ℝ

d

. In general, Riemann normal coordinates become singular when a

conjugate point

of

P

is encountered so they are typically more useful for studying local

geometry

than global geometry.

References

:
doCarmo 1992 (see

bibliography for differential geometry

)

Title

Riemann normal coordinates

Canonical name

RiemannNormalCoordinates

Date of creation

2013-03-22 14:35:35

Last modified on

2013-03-22 14:35:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53B05

Generated on Fri Feb 9 22:26:56 2018 by

LaTeXML
vector field along a curve

vector field along a curve

Let

M

be a

differentiable manifold

and

γ

:

[

a

,

b

]

→

M

be a

differentiable

curve in

M

. Then a vector field along

γ

is a differentiable map

Γ

:

[

a

,

b

]

→

T

⁢

M

, the

tangent bundle

of

M

, which projects to

γ

under the natural projection

π

:

T

⁢

M

→

M

. That is, it assigns to each point

t

0

∈

[

a

,

b

]

a vector

tangent

to

M

at the point

γ

⁢

(

t

)

, in a

continuous

manner. A good example of a vector field along a curve is the speed vector

γ

˙

. This is the

pushforward

of the constant vector field

d

d

⁢

t

by

γ

, i.e., at

t

0

, it is the derivation

γ

˙

⁢

(

f

)

=

d

d

⁢

t

⁢

(

f

∘

γ

)

|

t

=

t

0

.

Title

vector field along a curve

Canonical name

VectorFieldAlongACurve

Date of creation

2013-03-22 13:58:55

Last modified on

2013-03-22 13:58:55

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 53B05

Generated on Fri Feb 9 22:26:58 2018 by

LaTeXML
Riemann curvature tensor

Riemann curvature tensor

Let

𝒳

denote the

vector space

of smooth

vector fields

on a
smooth

Riemannian manifold

(

M

,

g

)

. Note that

𝒳

is actually a

𝒞

∞

⁢

(

M

)

module because we can multiply a vector field
by a function to obtain another vector field.
The

Riemann curvature tensor

is the tri-linear

𝒞

∞

mapping

R

:

𝒳

×

𝒳

×

𝒳

→

𝒳

,

which is defined by

R

⁢

(

X

,

Y

)

⁢

Z

=

∇

X

⁡

∇

Y

⁡

Z

-

∇

Y

⁡

∇

X

⁡

Z

-

∇

[

X

,

Y

]

⁡

Z

where

X

,

Y

,

Z

∈

𝒳

are vector fields, where

∇

is
the

Levi-Civita connection

attached to the

metric tensor

g

, and
where the square brackets denote the

Lie bracket

of two vector fields.
The tri-linearity means that for every smooth

f

:

M

→

ℝ

we have

f

⁢

R

⁢

(

X

,

Y

)

⁢

Z

=

R

⁢

(

f

⁢

X

,

Y

)

⁢

Z

=

R

⁢

(

X

,

f

⁢

Y

)

⁢

Z

=

R

⁢

(

X

,

Y

)

⁢

f

⁢

Z

.

In

components

this tensor is classically denoted by a set of
four-indexed components

R

i

j

⁢

k

⁢

l

. This means that given a
basis of

linearly independent

vector fields

X

i

we have

R

⁢

(

X

j

,

X

k

)

⁢

X

l

=

∑

s

R

s

j

⁢

k

⁢

l

⁢

X

s

.

In a two dimensional

manifold

it is known that the

Gaussian curvature

it is given by

K

g

=

R

1212

g

11

⁢

g

22

-

g

12

2

Title

Riemann curvature tensor

Canonical name

RiemannCurvatureTensor

Date of creation

2013-03-22 16:26:17

Last modified on

2013-03-22 16:26:17

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

10

Author

juanman (12619)

Entry type

Definition

Classification

msc 53B20

Classification

msc 53A55

Related topic

Curvature

Related topic

Connection

Related topic

FormalLogicsAndMetaMathematics

Generated on Fri Feb 9 22:27:08 2018 by

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codifferential

codifferential

The

codifferential

δ

of a

k

-form on an

n

-dimensional

Riemannian manifold

is given by:

(

-

1

)

n

⁢

(

k

+

1

)

+

1

∗

d

∗

where

∗

is the

Hodge star operator

and

d

is the

exterior
derivative

.

Let

g

denote the matrix locally representing the metric with
respect to co-ordinates

x

1

,

⋯

,

x

n

. Then for a 1-form

w

we
have:

δ

w

=

-

1

(

Det

⁢

g

)

∂

∂

⁡

x

i

[

√

(

Det

g

)

{

g

-

1

}

i

⁢

j

w

j

]

Title

codifferential

Canonical name

Codifferential

Date of creation

2013-03-22 18:37:11

Last modified on

2013-03-22 18:37:11

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

5

Author

whm22 (2009)

Entry type

Definition

Classification

msc 53B21

Related topic

DifferentialForms

Related topic

Laplacian

Generated on Fri Feb 9 22:27:15 2018 by

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Euclidean space as a manifold

Euclidean space as a manifold

Let

𝔼

n

be

n

-dimensional

Euclidean space

, and let

(

𝕍

,

⟨

⋅

,

⋅

⟩

)

be the corresponding

n

-dimensional
inner product space of

translation

isometries

. Alternatively, we can
consider Euclidean space as an inner product space that has forgotten
which point is its origin. Forgetting even more information, we have
the structure of

𝔼

n

as a

differential manifold

. We can obtain an
atlas with just one

coordinate chart

, a

Cartesian coordinate system

(

x

1

,

…

,

x

n

)

which gives us a

bijection

between

𝔼

n

and

ℝ

n

. The

tangent bundle

is trivial, with

T

⁡

𝔼

n

≅

𝔼

n

×

𝕍

.

Equivalently, every

tangent space

T

p

⁡

𝔼

n

,

p

∈

𝔼

n

. is

isomorphic

to

𝕍

.

We can retain a bit more structure, and consider

𝔼

n

as a

Riemannian
manifold

by equipping it with the

metric tensor

g

=

d

⁢

x

1

⊗

d

⁢

x

1

+

⋯

+

d

⁢

x

n

⊗

d

⁢

x

n

=

δ

i

⁢

j

⁢

d

⁢

x

i

⊗

d

⁢

x

j

.

We can also describe

g

in a coordinate-free fashion as

g

⁢

(

u

,

v

)

=

⟨

u

,

v

⟩

,

u

,

v

∈

𝕍

.

Properties

1.

Geodesics

are

straight lines

in

ℝ

n

.

2.

The

Christoffel symbols

vanish identically.

3.

The

Riemann curvature tensor

vanish identically.

Conversely, we can
characterize Eucldiean space as a

connected

,

complete

Riemannian
manifold with vanishing

curvature

and trivial

fundamental group

.

Title

Euclidean space as a

manifold

Canonical name

EuclideanSpaceAsAManifold

Date of creation

2013-03-22 15:29:48

Last modified on

2013-03-22 15:29:48

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Definition

Classification

msc 53B21

Classification

msc 53B20

Generated on Fri Feb 9 22:27:17 2018 by

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