Finsler Geometry: Riemannian foundations and relativistic ...3 As in Riemannian Geometry we want a...

Finsler Geometry: Riemannian foundations andrelativistic applications. Lecture 2: Anisotropic Calculus

Miguel Angel Javaloyes (University of Murcia)

Partially supported by Spanish MINECO/FEDER project reference MTM2015-65430-P andFundacion Seneca (Region de Murcia) project 19901/GERM/15

XXVIII International Fall Workshop on Geometry and PhysicsICMAT (Madrid), September 2-5, 2019

Geodesics, Flag curvature and Connections

1 Geodesics are the curves that (locally) minimize the length

2 The flag curvature is a measure of how geodesics get apart

3 As in Riemannian Geometry we want a distinguished connection tomake computations easier

4 We have to deal with a dependence on direction. For example, thefundamental tensor

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

gives a scalar product for every v ∈ TM \ 0.

5 this dependence can make Finsler computations a sea of tensors

6 Because of this, part of the Riemannian community has put FinslerGeometry aside

7 We will try to put some order

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

gv (u,w) :=1

∂t∂sF 2(v + tu + sw)|t=s=0,

Anisotropic tensors

Let us put A = TM \ 0. An anisotropic tensor T ∈ Trs(M,A) in a manifold

M is a family of maps of the form

Tv : Tπ(v)M∗ ×

r︷︸︸︷· · · Tπ(v)M

∗ × Tπ(v)M ×s︷︸︸︷· · · Tπ(v)M → R

for every v ∈ A such that in a system of coordinates (Ω, ϕ = (x1, . . . , xn)),the functions

T j1i2...jsi1i2...ir

(v) = Tv (dx i1 , dx i2 , . . . , dx ir , ∂j1 , ∂j2 . . . , ∂js )

are smooth.

One can identify T10(M,A) with the anisotropic vector fields X , namely,

smooth mapsA 3 v → X (v) ∈ TM

with X (v) ∈ Tπ(v)M.

Anisotropic tensors

Tv : Tπ(v)M∗ ×

r︷︸︸︷· · · Tπ(v)M

∗ × Tπ(v)M ×s︷︸︸︷· · · Tπ(v)M → R

are smooth.

Anisotropic tensors

Tv : Tπ(v)M∗ ×

r︷︸︸︷· · · Tπ(v)M

∗ × Tπ(v)M ×s︷︸︸︷· · · Tπ(v)M → R

are smooth.

Anisotropic connections

An anisotropic (linear) connection is a map

∇ : A×X(M)×X(M)→ TM, (v ,X ,Y ) 7→ ∇vXY := ∇(v ,X ,Y ) ∈ Tπ(v)M,

such that

(i) ∇vX (Y + Z ) = ∇v

XY +∇vXZ , for any X ,Y ,Z ∈ X(M) (linear),

(ii) ∇vX (fY ) = X (f )Y |π(v) + f (π(v))∇v

XY for any f ∈ F(M),X ,Y ∈ X(M) Leibniz rule,

(iii) ∇vfX+hYZ = f (π(v))∇v

XZ + h(π(v))∇vYZ , for any f , h ∈ F(M),

X ,Y ,Z ∈ X(M) (F(M)-linear),

(iv) for any X ,Y ∈ X(M), ∇XY ∈ T10(M,A) (considered as a map

A 3 v → ∇vXY ),

such that

(i) ∇vX (Y + Z ) = ∇v

A 3 v → ∇vXY ),

such that

(i) ∇vX (Y + Z ) = ∇v

A 3 v → ∇vXY ),

such that

(i) ∇vX (Y + Z ) = ∇v

A 3 v → ∇vXY ),

The torsion tensor of the connection is defined as

Tv (X ,Y ) = ∇vXY −∇v

YX − [X ,Y ]

Moreover, we will say that the anistropic linear connection is torsion-free ifT = 0, namely,

∇vXY −∇v

YX = [X ,Y ]

for every X ,Y ∈ X(M) and v ∈ A.

The torsion tensor of the connection is defined as

Tv (X ,Y ) = ∇vXY −∇v

YX − [X ,Y ]

Moreover, we will say that the anistropic linear connection is torsion-free ifT = 0, namely,

∇vXY −∇v

YX = [X ,Y ]

for every X ,Y ∈ X(M) and v ∈ A.

Christoffel symbols and auto-parallel curves

Given a system of coordinates (Ω, ϕ), we define theChristoffel symbols of ∇ as the functions

Γijk : A ∩ TΩ→ R

satisfying that

∇v∂j∂k = Γi

jk(v)∂i .Elwin B. Christoffel (1829-1900)

Observe that a connection is torsion-free if and only if its Christoffelsymbols are symmetric in j and k.

The auto-parallel curves of ∇ are the curves γ : [a, b]→ M such thatγ ∈ A and

γk + γ i γjΓkij(γ) = 0

We can define a covariant derivative Dγ along any curve γ and then

auto-parallel curves satisfy D γγ γ = 0

satisfying that

∇v∂j∂k = Γi

satisfying that

∇v∂j∂k = Γi

Semi-sprays and Sprays

Definition

Given a manifold M, a semi-spray on M is a vector field S in a subsetA ⊂ TM which has the following property:

if β : [a, b] ⊂ R→ A is an integral curve of S , then β = α, whereα = πM β.

We say that a semi-spray S is a spray if, in addition, A is conic, namely, ifv ∈ A, then λv ∈ A for every λ > 0 and S has the following property:

If β = α is an integral curve of S , then β(t) = λα(λt)is an integral curve of S for every λ > 0.

Semi-sprays and Sprays

Definition

Given a manifold M, a semi-spray on M is a vector field S in a subsetA ⊂ TM which has the following property:

if β : [a, b] ⊂ R→ A is an integral curve of S , then β = α, whereα = πM β.

We say that a semi-spray S is a spray if, in addition, A is conic, namely, ifv ∈ A, then λv ∈ A for every λ > 0 and S has the following property:

If β = α is an integral curve of S , then β(t) = λα(λt)is an integral curve of S for every λ > 0.

Berwald connection of a spray

In natural coordinates for TM, a vector field S in TM is a spray if and onlyif it is expressed as

S(v) =n∑

(y i∂

∂x i− 2G i (v)

∂y i)

with G i : TΩ ∩ A→ R positive homogeneous functions. Then

Γkij(v) =

∂Nkij

∂y j(v) =

∂2G k

∂y i∂y j(v)

are the Christoffel symbols of the Berwald connection ∇Auto-parallel curves of ∇ are the same as integral curves of S .Conclusion: Sprays can be studied as a particular case of anisotropicconnection

S(v) =n∑

(y i∂

∂x i− 2G i (v)

∂y i)

Γkij(v) =

∂Nkij

∂y j(v) =

∂2G k

∂y i∂y j(v)

S(v) =n∑

(y i∂

∂x i− 2G i (v)

∂y i)

Γkij(v) =

∂Nkij

∂y j(v) =

∂2G k

∂y i∂y j(v)

Deriving Anisotropic tensors with ∇First, we will see how to make the derivative of f : A→ R (A = TM \ 0)

Associated affine connection ∇V : Given an A-admissible vector field Vin an open subset Ω, we can define an affine connection ∇V using theanisotropic one:

(∇VXY )p = ∇V (p)

∇V is an affine connection in Ω ⊂ M.

Vertical derivation: given a tensor T ∈ Trs(M,A), we define its vertical

derivation as the tensor ∂νT ∈ Trs+1(M,A) given by

(∂νT )v (θ1, θ2, . . . , θr ,X1, . . . ,Xs ,Z )

∂tTv+tZ(π(v))(θ1, θ2, . . . , θr ,X1, . . . ,Xs)|t=0

for any

(θ1, . . . , θr ,X1, . . . ,Xs ,Z ) ∈ Tπ(v)M∗×

r︷︸︸︷· · · Tπ(v)M

∗×Tπ(v)M

s+1︷︸︸︷· · · Tπ(v)M

(∇VXY )p = ∇V (p)

(∂νT )v (θ1, θ2, . . . , θr ,X1, . . . ,Xs ,Z )

for any

(θ1, . . . , θr ,X1, . . . ,Xs ,Z ) ∈ Tπ(v)M∗×

r︷︸︸︷· · · Tπ(v)M

∗×Tπ(v)M

s+1︷︸︸︷· · · Tπ(v)M

(∇VXY )p = ∇V (p)

(∂νT )v (θ1, θ2, . . . , θr ,X1, . . . ,Xs ,Z )

for any

(θ1, . . . , θr ,X1, . . . ,Xs ,Z ) ∈ Tπ(v)M∗×

r︷︸︸︷· · · Tπ(v)M

∗×Tπ(v)M

s+1︷︸︸︷· · · Tπ(v)M

The subset of anisotropic functions is denoted by

F(A) = f : A = TM \ 0→ R

Given an anisotropic connection ∇ in A, we can define the derivation of afunction h ∈ F(A) as ∇Z (h) ∈ F(A) defined by

∇Z (h)(v) := Z (h(V ))(π(v))− (∂νh)v (∇vZV )

with V ,Z ∈ X(Ω), being V A-admissible on an open subset Ω ⊂ M suchthat V (π(v)) = v , which satisfies the Leibnitzian propertyD(fg) = D(f )g + fD(g) for any f , g ∈ F(A).

Proof: see Lemma 9 of

M. A. J., Anisotropic Tensor Calculus, 1941001, IJGMMP, 2019.In the following Jav19.

The subset of anisotropic functions is denoted by

F(A) = f : A = TM \ 0→ R

Given an anisotropic connection ∇ in A, we can define the derivation of afunction h ∈ F(A) as ∇Z (h) ∈ F(A) defined by

∇Z (h)(v) := Z (h(V ))(π(v))− (∂νh)v (∇vZV )

with V ,Z ∈ X(Ω), being V A-admissible on an open subset Ω ⊂ M suchthat V (π(v)) = v , which satisfies the Leibnitzian propertyD(fg) = D(f )g + fD(g) for any f , g ∈ F(A).

Proof: see Lemma 9 of

M. A. J., Anisotropic Tensor Calculus, 1941001, IJGMMP, 2019.In the following Jav19.

A tensor derivation is an R-linear map

D : ∪r≥0,s≥0Trs(M,A)→ ∪r≥0,s≥0T

rs(M,A)

that preserves the type of the tensor and satisfies the Leibnitz rule for thetensor product

D(T1 ⊗ T2) = D(T1)⊗ T2 + T1 ⊗ D(T2)

and it commutes with contractions

Theorem

Let ∇ be an anisotropic connection. Then there exists a unique tensorderivation ∇ such that (∇XY )(v) = ∇v

XY for every X ∈ X(M), and∇X (h) is defined as before.

Proof: see Theorem 11 of Jav19.

A tensor derivation is an R-linear map

D : ∪r≥0,s≥0Trs(M,A)→ ∪r≥0,s≥0T

rs(M,A)

that preserves the type of the tensor and satisfies the Leibnitz rule for thetensor product

D(T1 ⊗ T2) = D(T1)⊗ T2 + T1 ⊗ D(T2)

and it commutes with contractions

Theorem

Let ∇ be an anisotropic connection. Then there exists a unique tensorderivation ∇ such that (∇XY )(v) = ∇v

XY for every X ∈ X(M), and∇X (h) is defined as before.

Proof: see Theorem 11 of Jav19.

Derivative of an anisotropic tensor

By the product rule, if T ∈ T02(M,A), X ,Y ,Z ∈ X(M),

(∇ZT )v (X ,Y ) = ∇Z (T (X ,Y ))− Tv (∇ZX ,Y )− Tv (X ,∇ZY ).

By the Lemma, for any A-admissible extension V of v ,

∇Z (T (X ,Y ))(v) = Z (TV (X ,Y ))(π(v))− (∂νT )v (X ,Y ,∇VZ V )

Therefore,

(∇ZT )v (X ,Y ) = Z (TV (X ,Y ))(π(v))− (∂νT )v (X ,Y ,∇VZ V )

− Tv (∇VZ X ,Y )− Tv (X ,∇V

Z Y ).

Moral of the story: we can compute ∇ZT using the affine connection ∇V .

Therefore,

− Tv (∇VZ X ,Y )− Tv (X ,∇V

Z Y ).

Therefore,

− Tv (∇VZ X ,Y )− Tv (X ,∇V

Z Y ).

The curvature tensor of ∇

Using also the Lemma, we can make the derivative of an anisotropic vectorfield Y:

∇X (Y)(v) = ∇vX (Y(V ))− (∂νY)v (∇v

where (∂νY)v (z) = ddtY(v + tz)

∣∣t=0

, for any vector z ∈ Tπ(v)M.

Given X ,Y ,Z ∈ X(M), define

Rv (X ,Y )Z := ∇vX (∇YZ )−∇v

Y (∇XZ )−∇v[X ,Y ]Z

It is easy to prove that it is F(M)-linear in X ,Y ,Z .

Applying above formula, we get

∇vX (∇YZ ) = ∇v

X (∇VYZ )− ∂ν(∇YZ )v (∇v

∇X (Y)(v) = ∇vX (Y(V ))− (∂νY)v (∇v

∣∣t=0

Rv (X ,Y )Z := ∇vX (∇YZ )−∇v

Y (∇XZ )−∇v[X ,Y ]Z

∇vX (∇YZ ) = ∇v

X (∇VYZ )− ∂ν(∇YZ )v (∇v

∇X (Y)(v) = ∇vX (Y(V ))− (∂νY)v (∇v

∣∣t=0

Rv (X ,Y )Z := ∇vX (∇YZ )−∇v

Y (∇XZ )−∇v[X ,Y ]Z

∇vX (∇YZ ) = ∇v

X (∇VYZ )− ∂ν(∇YZ )v (∇v

Definition of the curvature tensor

Let us define “the vertical derivative” of the anisotropic connection ∇v as

Pv (X ,Y ,Z ) =∂

(∇v+tZ

X Y)|t=0,

Rv (X ,Y )Z := (∇VX∇V

YZ −∇VY∇V

XZ −∇V[X ,Y ]Z

− PV (Y ,Z ,∇VXV ) + PV (X ,Z ,∇V

YV ))(π(v)),

where V is any vector field with V (π(v)) = v , is well-defined and R is ananisotropic tensor.

Definition of the curvature tensor

Let us define “the vertical derivative” of the anisotropic connection ∇v as

Pv (X ,Y ,Z ) =∂

(∇v+tZ

X Y)|t=0,

Rv (X ,Y )Z := (∇VX∇V

YZ −∇VY∇V

XZ −∇V[X ,Y ]Z

− PV (Y ,Z ,∇VXV ) + PV (X ,Z ,∇V

YV ))(π(v)),

where V is any vector field with V (π(v)) = v , is well-defined and R is ananisotropic tensor.

Choosing V to simplify computations

Proposition

One can choose and A-admissible extension V defined in an open subsetΩ ⊂ M, such that

∇vXV = 0

for any vector field X ∈ X(Ω). Furthermore, if T ∈ Trs(M,A) and

X ∈ T10(M,A), then (∇XT )v = (∇V

X (TV ))π(v), and the curvature tensorof ∇ can be computed as

Rv (X ,Y )Z = RVπ(v)(X ,Y )Z = (∇V

X∇VYZ −∇V

Y∇VXZ )(π(v)),

for X ,Y ,Z ∈ X(Ω) such that [X ,Y ] = 0 (the last condition is notnecessary for the first identity), and its derivative as

(∇XR)v (Y ,Z )W = (∇VXR

V )π(v)(Y ,Z )W − Pv (Z ,W ,∇VX∇V

+ Pv (Y ,W ,∇VX∇V

Proposition

∇vXV = 0

X ∈ T10(M,A), then (∇XT )v = (∇V

Rv (X ,Y )Z = RVπ(v)(X ,Y )Z = (∇V

X∇VYZ −∇V

Y∇VXZ )(π(v)),

(∇XR)v (Y ,Z )W = (∇VXR

V )π(v)(Y ,Z )W − Pv (Z ,W ,∇VX∇V

Proposition

∇vXV = 0

X ∈ T10(M,A), then (∇XT )v = (∇V

Rv (X ,Y )Z = RVπ(v)(X ,Y )Z = (∇V

X∇VYZ −∇V

Y∇VXZ )(π(v)),

(∇XR)v (Y ,Z )W = (∇VXR

V )π(v)(Y ,Z )W − Pv (Z ,W ,∇VX∇V

Proposition

∇vXV = 0

X ∈ T10(M,A), then (∇XT )v = (∇V

Rv (X ,Y )Z = RVπ(v)(X ,Y )Z = (∇V

X∇VYZ −∇V

Y∇VXZ )(π(v)),

(∇XR)v (Y ,Z )W = (∇VXR

V )π(v)(Y ,Z )W − Pv (Z ,W ,∇VX∇V

Bianchi identities

As a consequence, we can get Bianchi Identities foranisotropic connections almost for free (using identities forthe affine connection ∇V ):

For every v ∈ A and u,w , z ∈ Tπ(v)M, we have thatRv (u,w) = −Rv (w , u) and R satisfies the 1st Bianchiidentity:∑cyc:u,w ,z

Rv (u,w)z =∑

cyc:u,w ,z

(Tv (Tv (u,w), z)+(∇uT )v (w , z)),

where T is the torsion of ∇, and the 2nd Bianchi identity:∑cyc:u,w ,z

((∇uR)v (w , z)b−Pv (w , b,Rv (u, z)v)+Rv (Tv (u,w), z)b

Here∑

cyc:u,w ,z denotes the cyclic sum in u,w , z .

Luigi Bianchi (1856-1928)

Bianchi identities

Rv (u,w)z =∑

cyc:u,w ,z

(Tv (Tv (u,w), z)+(∇uT )v (w , z)),

Here∑

Bianchi identities

Rv (u,w)z =∑

cyc:u,w ,z

(Tv (Tv (u,w), z)+(∇uT )v (w , z)),

Here∑

Vertical Bianchi identity

There is a new “vertical Bianchi identity”:

For every v ∈ A and u,w , z , b ∈ Tπ(v)M,

(∂νR)v (u,w , z , b) = (∇uP)v (w , z , b)− (∇wP)v (u, z , b)

− Pv (w , z ,Pv (u, v , b)) + Pv (u, z ,Pv (w , v , b))

+ Pv (Tv (u,w), z , b).

Vertical Bianchi identity

There is a new “vertical Bianchi identity”:

For every v ∈ A and u,w , z , b ∈ Tπ(v)M,

(∂νR)v (u,w , z , b) = (∇uP)v (w , z , b)− (∇wP)v (u, z , b)

− Pv (w , z ,Pv (u, v , b)) + Pv (u, z ,Pv (w , v , b))

+ Pv (Tv (u,w), z , b).

References

For the last results see Prop. 2.13, 2.14 and 2.15 of

M. A. J., Good connections and curvature computations in FinslerGeometry, arXiv:1904.07178 [math.DG].

In the following, Jav19b

Back to the Finslerian World

Derivation of the fundamental tensor

Let us recall that the fundamental tensor of a pseudo-Finsler metricL : A→ R is defined as

gv (u,w) :=1

∂t∂sL(v + tu + sw)|t=s=0,

and the Cartan tensor:

Cv (w1,w2,w3) :=1

∂s3∂s2∂s1L

3∑i=1

)∣∣∣∣∣s1=s2=s3=0

Then it follows that

(∂νg)v (w1,w2,w3) = 2Cv (w1,w2,w3)

(∇Zg)v (X ,Y ) := Z (gV (X ,Y ))− gV (∇VZ X ,Y )

− gV (X ,∇VZ Y )− 2CV (∇V

Z V ,X ,Y ),

Derivation of the fundamental tensor

Let us recall that the fundamental tensor of a pseudo-Finsler metricL : A→ R is defined as

gv (u,w) :=1

∂t∂sL(v + tu + sw)|t=s=0,

and the Cartan tensor:

Cv (w1,w2,w3) :=1

∂s3∂s2∂s1L

3∑i=1

)∣∣∣∣∣s1=s2=s3=0

Then it follows that

(∂νg)v (w1,w2,w3) = 2Cv (w1,w2,w3)

(∇Zg)v (X ,Y ) := Z (gV (X ,Y ))− gV (∇VZ X ,Y )

− gV (X ,∇VZ Y )− 2CV (∇V

Z V ,X ,Y ),

Chern connection as a Levi-Civita connection

It is well-known that when the Chern connection is interpreted as a familyof affine connections is characterized by satisfying:

XY −∇VYX = [X ,Y ] (torsion-free)

Z (gV (X ,Y )) = gV (∇VZ X ,Y )− gV (X ,∇V

Z Y )− 2CV (∇VZ V ,X ,Y )

But this is the same as ∇ being torsion-free and ∇g = 0 with ourdefinition.

The Chern connection is the Levi-Civita connection of a Finsler metric!!!!

Z Y )− 2CV (∇VZ V ,X ,Y )

Some Bibliography about this treatment

H.-H. Matthias, Zwei Verallgemeinerungen eines Satzes vonGromoll und Meyer, Bonner Mathematische Schriften [BonnMathematical Publications], 126, Universitat Bonn, MathematischesInstitut, Bonn, 1980.Dissertation, Rheinische Friedrich-Wilhelms-Universitat, Bonn, 1980.

H.-B. Rademacher, A sphere theorem for non-reversible Finslermetrics, Math. Ann., 328 (2004), pp. 373–387.

Z. Shen, Differential Geometry of Spray and Finsler Spaces, KluwerAcademic Publishers, Dordrecht, 2001.

The family of affine connections

the idea of the family of affine connections appears first related to theosculating metric gVthe osculating metric was considered in 1936 by A. Nazim in his PhDThesis

In 1941, O. Varga proves that when V is a geodesic vector field, thereis an affine connection related to the osculating metric having all theintegral curves of V as geodesics

In 1978, H.-H. Matthias introduces the concept of family of affineconnections related to the Chern connection when generalizing theGromoll-Meyer theorem

In 2001, Z. Shen includes this approach to Chern connection in hisbook about Sprays proving some results of Jacobi operator when V isgeodesic

In 2004, H.-B. Rademacher uses the family of affine connections tostudy the energy funcional in a sphere theorem for non-reversibleFinsler metrics

The classical connections

The classical connections used in Finsler Geometry are connections inthe vertical bundle π : π∗(TM)→ TM \ 0, whereπ∗(TM) = V(TM \ 0)

A connection in the vertical bundle is a map

∇′ : X(TM \ 0)× Γ(π∗(TM))→ Γ(π∗(TM))

Consider the natural coordinates of the tangent bundle ϕ = (x , y)associated with a system of coordinates (Ω, ϕ) in M, whereϕ−1(x1, . . . , xn, y1, . . . , yn) = y i ∂

∂x i

∣∣ϕ−1(x)

The Christoffel symbols are determined by ∇′ ∂∂xi

∂∂y j = Γk

ij(x , y) ∂∂yk

and ∇′ ∂∂yα

∂∂y j = Γk

αj(x , y) ∂∂yk

With anisotropic connections you need less information (there are halfof Christoffel symbols)

∇′ : X(TM \ 0)× Γ(π∗(TM))→ Γ(π∗(TM))

∂x i

∣∣ϕ−1(x)

∂∂y j = Γk

ij(x , y) ∂∂yk

∂∂y j = Γk

αj(x , y) ∂∂yk

∇′ : X(TM \ 0)× Γ(π∗(TM))→ Γ(π∗(TM))

∂x i

∣∣ϕ−1(x)

∂∂y j = Γk

ij(x , y) ∂∂yk

∂∂y j = Γk

αj(x , y) ∂∂yk

∇′ : X(TM \ 0)× Γ(π∗(TM))→ Γ(π∗(TM))

∂x i

∣∣ϕ−1(x)

∂∂y j = Γk

ij(x , y) ∂∂yk

∂∂y j = Γk

αj(x , y) ∂∂yk

∇′ : X(TM \ 0)× Γ(π∗(TM))→ Γ(π∗(TM))

∂x i

∣∣ϕ−1(x)

∂∂y j = Γk

ij(x , y) ∂∂yk

∂∂y j = Γk

αj(x , y) ∂∂yk

Anisotropic connections associated with the classicalconnections

A Finsler metric has always associated a spray (constructed from itsgeodesics), and then an Ehresmann connection, which allows one to lift avector field X ∈ X(M) to a vector field XH ∈ X(TM). Moreover, using

iv : Tπ(v)M → VvTM

given by iv (u) = ddt (v + tu)|t=0, we also can lift a vector field X ∈ X(M)

to a section of π : π∗(TM)→ TM \ 0.

Anisotropic connections associated with the classicalconnections

A Finsler metric has always associated a spray (constructed from itsgeodesics), and then an Ehresmann connection, which allows one to lift avector field X ∈ X(M) to a vector field XH ∈ X(TM). Moreover, using

iv : Tπ(v)M → VvTM

given by iv (u) = ddt (v + tu)|t=0, we also can lift a vector field X ∈ X(M)

to a section of π : π∗(TM)→ TM \ 0.

Recall that, we can define an anisotropic connection

∇vXY = (∇XHY

V)(v),

and an anisotropic tensor

Cv (X ,Y ) = (∇XVYV)(v),

Then for classical connections it turns out that

Chern-Rund connection projects into the Levi-Civita and C = 0

Cartan connection projects into Levi-Civita and C = C [

Berwald connection projects into Berwald and C = 0

Hashiguchi connection projects into Berwald and C = C [

For details see §4.4 of Jav19b

∇vXY = (∇XHY

V)(v),

Cv (X ,Y ) = (∇XVYV)(v),

∇vXY = (∇XHY

V)(v),

Cv (X ,Y ) = (∇XVYV)(v),

∇vXY = (∇XHY

V)(v),

Cv (X ,Y ) = (∇XVYV)(v),

∇vXY = (∇XHY

V)(v),

Cv (X ,Y ) = (∇XVYV)(v),

∇vXY = (∇XHY

V)(v),

Cv (X ,Y ) = (∇XVYV)(v),

Berwald and Chern connections

Let ∇ be the Berwald connection. The Berwald tensor B is the verticalderivative of ∇ and the Landsberg curvature

Lv (u,w , z) = 2gv (Bv (u,w , z), v)

If we define L[ as gv (L[v (u,w), z) = Lv (u,w , z), we can write down thedifference tensor between the Chern and Berwald connections as

∇vXY − ∇v

XY = L[v (X ,Y ),

see (7.17) in

As a consequence if P is the vertical derivative of the Chern connection:Pv (v , v , u) = 0, for every v ∈ A and u,w ∈ Tπ(v)M (see Prop. 3.6 ofJav19b)

Lv (u,w , z) = 2gv (Bv (u,w , z), v)

∇vXY − ∇v

XY = L[v (X ,Y ),

see (7.17) in

Lv (u,w , z) = 2gv (Bv (u,w , z), v)

∇vXY − ∇v

XY = L[v (X ,Y ),

see (7.17) in

Observe that L[v (v , u) = 0. It turns out that all the torsion-freeconnections such that

∇vXY − ∇v

XY = Qv (X ,Y ),

with Qv (v , u) = 0, are good connections to study Finsler Geometry.

Properties of the curvature tensor

Proposition

Let (M, L) be a pseudo-Finsler manifold and ∇, its Levi-Civita-Chernconnection. Then the curvature tensor R associated with ∇ satisfies thesymmetries:

gv (Rv (u,w)z , b) + gv (Rv (u,w)b, z) = 2Cv (Rv (w , u)v , z , b)

gv (Rv (u,w)z , b)− gv (Rv (z , b)u,w) =

Cv (Rv (w , z)v , u, b) + Cv (Rv (z , u)v ,w , b) + Cv (Rv (u, b)v , z ,w)

+ Cv (Rv (b,w)v , z , u) + Cv (Rv (z , b)v , u,w) + Cv (Rv (w , u)v , z , b).

For a proof see Prop. 3.1 of Jav19b.

Geodesics locally minimize

Proposition

Let (M,F ) be a Finsler metric. Then there exists r > 0 such that

expp : B+0 (r)→ expp(B+

0 (r))

is a diffeomorphism, and in this case, for any q ∈ B+p (r) the radial

geodesic from p to q is, up to reparametrizations, the unique minimizer ofthe Finslerian distance.

Proof:

given any other curve c from p to q, c(u) = expp(s(u)v(u)), withv : [0, 1]→ Σp, s(0) = 0, s(1) = r

σ(t, u) = expp(tv(u)), T (t, u) = ∂σ∂t (t, u),

U(t, u) =∂σ

∂u(t, u) = d expp(tv(u))

c(u) = σ(s(u), u), c(u) = s(u)T (s(u), u) + U(s(u), u)

By the Fund. Ineq. gT (T , c) ≤ F (T )F (c), and by the Gauss LemmagT (T ,U) = 0

F (T )F (c) ≥ gT (T , c) = s(u)gT (T ,T ) and then s(u) ≤ F (c)(F (T ) = 1)

`F (c) =∫ 1

0 F (c)du ≥∫ 1

0 s(u)du = s(1)− s(0) = r .

Proof:

U(t, u) =∂σ

`F (c) =∫ 1

0 F (c)du ≥∫ 1

0 s(u)du = s(1)− s(0) = r .

Proof:

U(t, u) =∂σ

`F (c) =∫ 1

0 F (c)du ≥∫ 1

0 s(u)du = s(1)− s(0) = r .

Proof:

U(t, u) =∂σ

`F (c) =∫ 1

0 F (c)du ≥∫ 1

0 s(u)du = s(1)− s(0) = r .

Proof:

U(t, u) =∂σ

`F (c) =∫ 1

0 F (c)du ≥∫ 1

0 s(u)du = s(1)− s(0) = r .

Proof:

U(t, u) =∂σ

`F (c) =∫ 1

0 F (c)du ≥∫ 1

0 s(u)du = s(1)− s(0) = r .

Variation of the energy

For any piecewise smooth curve γ : [a, b] ⊂ R→ M, let us define theenergy functional as

E (γ) =1

aF (γ)2ds. (1)

The second fundamental form of P in the direction of N as the tensor

SPN : X(P)× X(P)→ X(P)⊥N

given by SPN (U,W ) = norN∇NUW , where norN is computed with the

metric gN , and X(P)⊥N is the space of gN -orthogonal vector fields to P.

Variation of the energy

For any piecewise smooth curve γ : [a, b] ⊂ R→ M, let us define theenergy functional as

E (γ) =1

aF (γ)2ds. (1)

The second fundamental form of P in the direction of N as the tensor

SPN : X(P)× X(P)→ X(P)⊥N

given by SPN (U,W ) = norN∇NUW , where norN is computed with the

metric gN , and X(P)⊥N is the space of gN -orthogonal vector fields to P.

Second variation and Flag curvature

E ′(0) :=d(E (γs))

ds|s=0

= −∫ b

agγ(W ,D γ

γ γ) dt + gγ(W , γ)|ba

(LL(γ(t+

i ))(W (ti ))− LL(γ(t−i ))(W (ti ))),

where LL(v)(w) = gv (v ,w) is the Legendre transform and D theLevi-Civita connection. Moreover, if γ is a geodesic which is gγ-orthogonalto two submanifolds P and Q at the endpoints, then

E ′′(0) =

(−gγ(Rγ(γ,W )W , γ) + gγ(D γ

γW ,D γγW )

+ gγ(b)(SPγ(b)(W ,W ), γ(b))− gγ(a)(SQγ(a)(W ,W ), γ(a)),

Second variation and Flag curvature

E ′(0) :=d(E (γs))

ds|s=0

= −∫ b

agγ(W ,D γ

γ γ) dt + gγ(W , γ)|ba

(LL(γ(t+

i ))(W (ti ))− LL(γ(t−i ))(W (ti ))),

where LL(v)(w) = gv (v ,w) is the Legendre transform and D theLevi-Civita connection. Moreover, if γ is a geodesic which is gγ-orthogonalto two submanifolds P and Q at the endpoints, then

E ′′(0) =

(−gγ(Rγ(γ,W )W , γ) + gγ(D γ

γW ,D γγW )

+ gγ(b)(SPγ(b)(W ,W ), γ(b))− gγ(a)(SQγ(a)(W ,W ), γ(a)),

Morse Theory of the energy functional

The main difficulty is that the square of a Finsler metric F 2 is smoothin the zero section if and only if it is a Riemannian metric.

As a consequence, the energy functional is not C 2 in the space ofcurves with regularity H1.

there are two possibilities: to consider the space of broken geodesicsor to proceed as in the reference, providing a splitting lemma for theFinslerian energy functional

E. Caponio, M. A. Javaloyes, and A. Masiello.Morse theory of causal geodesics in a stationary spacetime via Morsetheory of geodesics of a Finsler metric.Ann. Inst. H. Poincare Anal. Non Lineaire, 27(3):857–876, 2010.

Flag Curvature

the flag curvature of F is given by:

K (v ,w) =gv (Rv (v ,w)w , v)

F (v)2gv (w ,w)− gv (v ,w)2.

It measures how geodesics taking initial values in the planeπ = spanv ,w get apart from the geodesic γ such that γ(0) = v

Flag Curvature

the flag curvature of F is given by:

K (v ,w) =gv (Rv (v ,w)w , v)

F (v)2gv (w ,w)− gv (v ,w)2.

It measures how geodesics taking initial values in the planeπ = spanv ,w get apart from the geodesic γ such that γ(0) = v

Interpretation of the Flag Curvature

See §5.5 of:

Bao, Chern, Shen, An introduction to Riemann-Finsler Geometry,Springer, 2001.

Constant flag curvature

There is no classification of Finsler manifolds with constant flagcurvature

The only known case is the family of Randers metrics

In this case, a Randers metrics has constant flag curvature if and onlyif its Zermelo data (h,W ) satisfies that h has constant curvature andW is a homothety. This was proved in the reference:

D. Bao, C. Robles, and Z. Shen, Zermelo navigation onRiemannian manifolds, J. Differential Geom., 66 (2004), pp. 377–435.

Submanifolds and orthogonal geodesics

Geodesics γ : [a, b]→ M (Critical points of E ) are auto-parallelcurves of the Chern connection orthogonal to P and Q, namely,gγ(a)(γ(a), u) = 0 for all u ∈ Tγ(a)P and gγ(b)(γ(b),w) = 0 for allw ∈ Tγ(b)Q.

They are also critical points of the length functional.

Do they realize locally the distance to a submanifold P?

dF (p,P) = infγ∈C(p,P)

aF (γ)ds

where C (p,P) = γ : [a, b]→ M : γ(a) = p, γ(b) ∈ P.the answer is affirmative

aF (γ)ds

Benigno Alves and M. A. J., A note on the existence of tubularneighbourhoods on Finsler manifolds and minimization of orthogonalgeodesics to a submanifold, PAMS, Vol. 147, 369-376 (2019).

The main difficulty to generalize this result to Finsler Geometry isthat the orthogonal space to P, ν(P), is not a vector bundle (it is notlinear)

ν(P) = v ∈ TM : π(v) ∈ P, gv (v , u) = 0 ∀u ∈ Tπ(v)P

ν(P) is a cone.

Then one cannot use the classical result of existence of tubularneighbourhoods for vector bundles

Benigno Alves and M. A. J., A note on the existence of tubularneighbourhoods on Finsler manifolds and minimization of orthogonalgeodesics to a submanifold, PAMS, Vol. 147, 369-376 (2019).

The main difficulty to generalize this result to Finsler Geometry isthat the orthogonal space to P, ν(P), is not a vector bundle (it is notlinear)

ν(P) = v ∈ TM : π(v) ∈ P, gv (v , u) = 0 ∀u ∈ Tπ(v)P

ν(P) is a cone.

Then one cannot use the classical result of existence of tubularneighbourhoods for vector bundles

Strategy of the proof

When P is a hypersurface, the orthogonal space provides two vectorbundles

This allows one to get the result

When the codimension of P is bigger than one, for every v ∈ ν(P):

it is possible to find a hypersurface Pv ⊃ P and v ∈ ν(Pv )

If the geodesic γv minimizes the distance to Pv , it also minimizes thedistance to P

Existence of Tubular neighbourhoods

Theorem

For any precompact open subset Q of P, there exists ε > 0 such that theexponential map is defined in the open subset

Vε = v ∈ ν(Q) : F (v) < ε,

exp : Vε → Vε \ Q is a diffeomorphism for a certain open subset Vε ⊂ M

and the geodesic γv : [0, 1]→ M minimizes the distance from P to γv (1)for all v ∈ Vε.

In particular, exp(Vε) is a tubular neighbourhood of Q.

Proof:

making the choice of Pv with smooth dependence on v one can find acommon interval of minimization for a neighbourhood of v in ν(P).

Theorem

Vε = v ∈ ν(Q) : F (v) < ε,

Proof:

Theorem

Vε = v ∈ ν(Q) : F (v) < ε,

Proof:

Theorem

Vε = v ∈ ν(Q) : F (v) < ε,

Proof:

Lie derivative

For every X ∈ X(M), we define

LX (L) = X (L(V ))− (∂νL)v ([X ,V ]),

where V is any vector field that extends v ∈ A. Now observe that(∂νL)v (w) = 2gv (v ,w). Then using the Chern connection and thatL(v) = gv (v , v) we get

LX (L) = 2gV (∇VXV ,V )− 2gV (V , [X ,V ]) = 2gv (∇v

vX , v).

It follows that X is a Killing field of L if and only if LXL = 0 andconformal if and only if LXL = fL for some function f : M → R

Lie derivative

LX (L) = X (L(V ))− (∂νL)v ([X ,V ]),

vX , v).

Lie derivative

LX (L) = X (L(V ))− (∂νL)v ([X ,V ]),

vX , v).

Lie derivative

Moreover,

(LXg)v (Y ,Z ) = X (gV (Y ,Z ))− gV ([X ,Y ],Z )

− gV (Y , [X ,Z ])− 2CV ([X ,V ],Y ,Z )

since ∂νg = 2C . Using the Chern connection we get the tensorialexpression

(LXg)v (u,w) = gv (∇vuX ,w) + gv (u,∇v

wX )− 2Cv (∇vvX , u,w),

which gives another characterization of Killing fields:

gv (∇vuX ,w) + gv (u,∇v

wX )− 2Cv (∇vvX , u,w) = 0,

for every v ∈ A and u,w ∈ Tπ(v)M.

Lie derivative

Moreover,

− gV (Y , [X ,Z ])− 2CV ([X ,V ],Y ,Z )

wX )− 2Cv (∇vvX , u,w),

wX )− 2Cv (∇vvX , u,w) = 0,

Lie derivative

Moreover,

− gV (Y , [X ,Z ])− 2CV ([X ,V ],Y ,Z )

wX )− 2Cv (∇vvX , u,w),

wX )− 2Cv (∇vvX , u,w) = 0,

Lie derivative

Let us observe that given a diffeomorphism ψ : M → M, we can define thepullback ψ∗(T ) of an anisotropic tensor T as the anisotropic tensor givenby ψ∗(T )v = Tψ∗(v), where ψ∗ is the differential of ψ.

Proposition

If X ∈ X(M) and T ∈ T0s (M,A), then

LXT = limt→0

t(ψ∗t (T )− T ),

where ψt is the (possibly local) flow of X .

For the proof see Prop. 28 of Jav19.

Lie derivative

Let us observe that given a diffeomorphism ψ : M → M, we can define thepullback ψ∗(T ) of an anisotropic tensor T as the anisotropic tensor givenby ψ∗(T )v = Tψ∗(v), where ψ∗ is the differential of ψ.

Proposition

If X ∈ X(M) and T ∈ T0s (M,A), then

LXT = limt→0

t(ψ∗t (T )− T ),

where ψt is the (possibly local) flow of X .

For the proof see Prop. 28 of Jav19.

Main References

Recall that Jav19 is

M. A. J., Anisotropic Tensor Calculus, 1941001, IJGMMP, 2019.and Jav19b is

M. A. J., Good connections and curvature computations in FinslerGeometry, arXiv:1904.07178 [math.DG].

Finsler Geometry: Riemannian foundations and relativistic ...3 As in Riemannian Geometry we want a...

Documents

Transcript of Finsler Geometry: Riemannian foundations and relativistic ...3 As in Riemannian Geometry we want a...