Yardsticks to Metric Tensor Fields

I requested myself why completely different scientists perceive the identical factor seemingly otherwise, particularly the idea of a metric tensor. If we ask a topologist, a classical geometer, an algebraist, a differential geometer, and a physicist “What’s a metric?” then we get 5 completely different solutions. I imply it’s all about distances, isn’t it? “Sure” continues to be the reply and all do truly imply the identical factor. It’s their perspective that’s completely different. This text is meant to elucidate how.

The picture exhibits medieval requirements of comparability at a church in Regensburg, Germany, Schuh (shoe), Elle (ulna), and Klafter.

Topology

The topologist has most likely the simplest idea. A metric is a property of sure topological areas, metric areas. It implies that the topological house is provided with a yardstick. A perform that measures distances. The circumstances are clear:

Symmetry

We don’t trouble whether or not we measure from proper to left or from left to proper.

Constructive Definiteness

The gap is rarely detrimental, and nil if and provided that there isn’t any distance between our factors, i.e. it’s the similar level. Sounds foolish, however there are certainly unique topological areas by which this isn’t a triviality. We simply don’t need to trouble about these pathological examples.

Triangle Inequality

If we’re going from one level to a different and making a cease at a 3rd level that isn’t in our direct method, then it’s a detour.

All people will perceive that this can be a passable description of what our yardstick is meant to do.

Classical Geometry

Not so the classical geometer! He doesn’t care about distances. His enterprise is comparisons. No, not shorter or longer. His view is how typically one distance suits into one other distance. One may argue that that is nonetheless the idea of a yardstick: how typically does a yard match right into a distance? Nevertheless, the distinction is delicate. The classical geometer doesn’t need to outline a unit, be it a yard or a meter. He all the time compares any two lengths and tells us the quotient of them, e.g. the golden ratio is outlined by
$$
dfrac{a+b}{a}=dfrac{a}{b}quadLongleftrightarrowquad a^2-b^2=ab.
$$
The traditional Greeks didn’t solely take care of distances. If we have a look at the trigonometric features, particularly quotients of two lengths once more, we get to the opposite necessary measure in classical geometry: angles and therewith triangles, ideally proper ones. We now have to contemplate instructions, too.

Angles are clearly associated to the multiplication of vectors, and that calls for two different necessary circumstances, the distributive legal guidelines.

Additions

##langle vec{a}+vec{b},vec{c}rangle=langle vec{a},vec{c} rangle +langle vec{b},vec{c} rangle## and ##langle vec{a},vec{b}+vec{c} rangle =langle vec{a},vec{b} rangle +langle vec{a},vec{c} rangle##

Stretches and Compressions

i.e. Scalar Multiples

## alpha langle vec{a},vec{b} rangle =langle alpha vec{ a},vec{b} rangle= langle vec{a},alpha vec{b} rangle##

If we now outline a size by
$$
|vec{a}| :=sqrt{langle vec{a},vec{a} rangle}=sqrt{sum_{ok=1}^n a_k^2},
$$
we get a metric within the topological sense
$$
d(vec{a},vec{b}):=|vec{a}-vec{b}|=sqrt{sum_{ok=1}^n (a_k-b_k)^2}.
$$
and an angle
$$
cos(sphericalangle(vec{a},vec{b}))=dfrac{langle vec{a},vec{b} rangle}{|vec{a}|cdot |vec{b}|}= dfrac{sum_{ok=1}^na_kb_k}{sqrt{sum_{ok=1}^n a_k^2}sqrt{sum_{ok=1}^n b_k^2}}.
$$

Summary Algebra

An algebraist is within the buildings, not a lot in coordinates, so we outline a metric tensor by the properties above somewhat than coordinates. A metric tensor for a vector house ##V## is due to this fact merely a component of ##V^*otimes V^*.## Effectively, that is true, quick, and doesn’t assist us in any respect. What we need to outline is an internal product ##langle , . ,, , ,, . , rangle .## That, at its core, is a symmetric, constructive particular, bilinear mapping
$$
g, : ,V instances V longrightarrow mathbb{R}
$$
Such a mapping will be written as a matrix ##G,## i.e. ##g(vec{a},vec{b})=vec{a}cdot G cdotvec{b}.## We now have thought-about ##g(vec{a},vec{b})=vec{a}cdot operatorname{Id}cdotvec{b}## to date, the place ##operatorname{Id}## is the id matrix. Nevertheless, there isn’t any purpose to limit ourselves to the id matrix. Any constructive particular matrix ##G## will do. That is mainly saying ##gin V^*otimes V^*.## Afterward, we are going to even drop this situation and solely demand that ##G## is non-degenerate, i.e. that for each vector ##vec{a}neq vec{0}## there’s a vector ##vec{b}## such that ##g(vec{a},vec{b})neq vec{0}.## With ##G=operatorname{Id}## we’ve got a selected instance of a vector house ##V=mathbb{R}^n## that provides the development a that means.

The metric tensor ##g_pin V^*otimes V^*## above an affine level house ##A={p}+V## with translation house ##V## is a mapping that assigns to ##{p}## a symmetric, constructive particular, bilinear mapping ##g_p.## Size, distance and angle for ##vec{a},vec{b}in V ## are actually given as

start{align*}
|vec{a}|_p&=sqrt{g_p(vec{a},vec{a})}
d_p(vec{a},vec{b})&=|vec{a}-vec{b}|_p
cos(sphericalangle_p(vec{a},vec{b}))&=dfrac{g_p(vec{a},vec{b})}{|vec{a}|_pcdot|vec{b}|_p}
finish{align*}
If we’ve got an affine linear monomorphism ##varphi , : ,{p}+U rightarrowtail {varphi(p)}+V## then any metric tensor on ##{varphi(p)}+V## defines a metric tensor on ##{p}+U## by the definition

$$(varphi^* g)_p(vec{u}_1,vec{u}_2):=g_{varphi(p)}(varphi_*(vec{u}_1),varphi_*(vec{u}_2)).$$

This pullback, or likewise the truth that ##g_pin T(V^*),## is the rationale the algebraist calls the metric tensor contravariant: Given one other monomorphism ##psi , : ,{varphi(p)} +V rightarrowtail {psi(varphi(p))}+W## then ##(psi circvarphi)^*(g_p)=g_{varphi(p)}circ g_{psi(varphi(p))}=(varphi^*circ psi^*)(g_p).## This alteration of order means for mathematicians contravariance. (Consideration: physicists name it covariant for various causes, see beneath!)

It is very important discover that each one three portions rely upon the purpose ##p.##

Manifolds

The entire above has been somewhat theoretical to date. Positive, the topologist may clarify our yardstick and the traditional Greeks had been actually wonderful in Euclidean geometry. However it’s not till we meet the differential geometer and the physicist to inform us one thing about the true world, the non-Euclidean realities, and the truth that the algebraist’s perspective was not utterly irrelevant. They deal e.g. with saddles:

If we improve the decision, rotate the saddle a bit, and eventually zoom in, and focus on that small marked sq.

then we are able to fake that this can be a small flat airplane that represents the tangent house in some unspecified time in the future inside the sq.. The smaller we select our sq.. the smaller the error in any calculations as a result of curvature. After which we are able to do the identical with any neighboring sq.. We solely demand that the calculations of any overlapping components must be the identical, for brief: charts, an atlas, and an internal product on ##T_pM##.

That’s it. We simply have understood the idea of Riemannian manifolds the place its native coordinates inside the sq. are the parts of foundation tangent vectors of the tangent house at a sure level inside the sq.. Since there isn’t any globally flat coordinate system on the saddle, we’ve got to patch all of the tiny flat squares, our charts, and bind them to an atlas. That’s what the differential geometer does. Observe the massive distinction between a world view (left) and the native view (proper).

One final, however essential be aware: our manifold right here, the saddle, is given by ##M={(x,y,x^2-y^2),|,x,yin mathbb{R}}.## This implies we’ve got two impartial parameters, ##x## and ##y,## so ##dim M=2,## a floor. And though we’ve got embedded it in our three-dimensional house, we won’t care about this embedding, solely concerning the factors on ##M.## The one purpose for an embedding is, that we are able to have fancy photos. Individuals on earth don’t acknowledge its curvature until they have a look at the horizon at sea, huge grass savannas, or from outer house.

Physics

Earlier than we hearken to the differential geometer who will inform us one thing about sections and assist clarify the algebraist, allow us to hear what the physicist has to say. Unexpectedly we discover ourselves in a well-known state of affairs.

The affine house is solely ##{p}+V=T_pM,## the tangent house on the level ##p## of the manifold ##M## outlined by the saddle perform ##z=x^2-y^2.## The metric tensor assigns a bilinear kind to ##T_pM.## So, relying on the place we’re, we’ve got size, distance, and angle. However physics is all about measurement. How a lot differs one thing if one thing else occurs? In scientific this implies: We want coordinates! And since we already understood the Riemannian manifold ##M,## they are going to be native coordinates; solely legitimate on ##T_pM## and permitting us an inexpensive measurement: symmetric, bilinear (distribution legal guidelines), and non-degenerate (to permit relativity idea) somewhat than constructive definiteness.

We now have seen that all of it relies upon (easily) on the placement ##p,## which is not any shock since differentiation is an area course of. However we are able to patch all native views to an atlas. This is the reason we communicate of fields as a substitute of areas. A subject is an object obtained by gathering all factors ##pin M.## Due to this fact we’ve got a metric tensor subject, and a tangent vector subject. It’s normally merely known as a vector subject, whereas the vector house of tangents known as tangent house.
start{align*}
d_p(vec{x}_1,vec{x}_2)=sqrt{g_p(vec{x}_1-vec{x}_2,vec{x}_1-vec{x}_2)}&quadtext{ metric}
g_p(vec{x}_1,vec{x}_2)=D_pvec{x}_1otimes D_pvec{x}_2&quadtext{ metric tensor}
g=bigsqcup_{pin M}g_p=bigcup_{pin M}{p}instances g_p&quadtext{ metric tensor subject}
T_pM&quadtext{ tangent (vector) house}
TM=bigsqcup_{pin M}T_pM=bigcup_{pin M}{p}instances T_pM&quadtext{ (tangent) vector subject}
finish{align*}

Coordinates

Let’s take into account our saddle ##M={(x,y,x^2-y^2):x,yin mathbb{R}}.## A tangent is the speed vector of a curve ##gamma :[0,1]rightarrow M .## We take into account ##gamma^1(t)=(t,p_y,t^2-p_y^2)## and ##gamma^2(t)=(p_x,t,p_x^2-t^2)## by some extent ##p=(p_x,p_y,p_z)in M## and get the tangents ##dotgamma^1(t)=(1,0,2t)## and ##dotgamma^2(t)=(0,1,-2t).## They correspond to the vectors fields ##X_1(p)=D_p(x)=(1,0,2p_x)## and ##X_2(p)=D_p(y)=(0,1,-2p_y).##

When it comes to degree units, we’ve got ##f(x,y)=x^2-y^2## and ##(p_x,p_y)in f^{-1}(p_x^2-p_y^2).## Then ##nabla f(p)=(2p_x,-2p_y).## Let additional be ##x,y:[-1,1]rightarrow f^{-1}(p_x^2-p_y^2)## outlined by ##x(t)=(t+p_x,sqrt{(t+p_x)^2-(p_x^2-p_y^2)}),x(0)=(p_x,p_y)## and ##y(t)=(sqrt{(t+p_y)^2+(p_x^2-p_y^2)},t+p_y),y(0)=(p_x,p_y).##
Then
start{align*}
nabla(f(x(0)))cdot dot x(0)&=(2p_x,-2p_y)cdot (1,p_x/p_y)=0
nabla(f(x(0)))cdot dot y(0)&=(2p_x,-2p_y)cdot (p_y/p_x,1)=0
finish{align*}
and the tangent house to the extent set in ##mathbb{R}^2## is ##left{nabla f(p)proper}^perp = operatorname{lin}_mathbb{R}left{(p_y,p_x)proper}##. Thus ##(p_y,p_x,0)=p_yX_1(p)+p_xX_2(p)## is one tangent vector to ##M## at ##p## within the ##z##-plane.

The tangent house with origin in ##p=(2,1,3)## is
$$
T_pM=
operatorname{lin}_mathbb{R}left{X_1(p)=left.dfrac{partial }{partial p_x}proper|_p,X_2(p)=left.dfrac{partial }{partial p_y}proper|_pright}=operatorname{lin_mathbb{R}}left{start{pmatrix}14end{pmatrix},start{pmatrix}01-2end{pmatrix} proper}
$$
and the vector subject is
$$
TM = bigsqcup_{pin M}operatorname{lin}_mathbb{R}left{X_1=dfrac{partial }{partial p_x},X_2=dfrac{partial }{partial p_y}proper}=bigsqcup_{pin M}operatorname{lin}_mathbb{R}left{start{pmatrix}12p_xend{pmatrix},start{pmatrix}01-2p_yend{pmatrix} proper}.
$$
Nevertheless, we don’t need to trouble with the embedding in ##mathbb{R}^3.## The entire thought about manifolds is, that they don’t require an embedding. There is no such thing as a Euclidean house across the universe, solely the universe itself. Effectively, we begin a bit smaller and see what the saddle will inform us. However who is aware of, perhaps the universe is a saddle, say a hyperbolic paraboloid. The metric tensor wants vectors as inputs. Our vectors are the tangent vectors ##{X_1(p),X_2(p)}## which kind a foundation for ##V=T_pM.## The affine house ##A## we’ve got spoken about within the algebra part is solely ##A={p}+T_pM.## These areas are all two-dimensional, so ##g_p## may have 4 elements. In an effort to illustrate the dependency on location, we outline based on the body ##mathbf{f}={X_1(p),X_2(p)}##
start{align*}
g_p&triangleq G[mathbf{f}]=g_{ij}[mathbf{f}]=start{pmatrix}
cosh^2 p_x +sinh^2 p_y &0&(1+4p_x^2+4p_y^2)^{-1}
finish{pmatrix}.
finish{align*}
If we alter the coordinates in ##T_pM## by
##
U_k=(A_{ij})cdot X_k=sum_{l}a_{lk}X_l={a^l}_kX_l
##
then (altering the body)
start{align*}
g(U_i,U_j)&= g({a^ok}_i X_k,{a^l}_jX_l)={a^ok}_i{a^l}_jg(X_k,X_l)
&={a^ok}_i{a^l}_jg_{kl}[mathbf{f}]=sum_{ok,l}(A_{ki})g_{kl}(A_{lj})[mathbf{f}]=left(A^tau cdot G[mathbf{f}]cdot Aright)_{ij}
finish{align*}
and ##G[Amathbf{f}]=A^tau G[mathbf{f}]A,## i.e. the metric tensor, or somewhat its coordinate illustration, known as covariant in physics since its elements rework like the premise underneath a change of coordinates in every index of the body.

If we alter the coordinate features ##p_x,p_y:Mrightarrow mathbb{R}## to ##q_x,q_y:Mrightarrow mathbb{R}## then
$$
dfrac{partial }{partial q_x}=dfrac{partial p_x}{partial q_x}dfrac{partial }{partial p_x}+dfrac{partial p_y}{partial q_x}dfrac{partial }{partial p_y}; , ;dfrac{partial }{partial q_y}=dfrac{partial p_x}{partial q_y}dfrac{partial }{partial p_x}+dfrac{partial p_y}{partial q_y}dfrac{partial }{partial p_y}
$$
and with the Jacobi matrix ##Dq(p)##
start{align*}
g_{q(p)}&=gleft(dfrac{partial }{partial q_x},dfrac{partial }{partial q_y}proper)=gleft(dfrac{partial p_x}{partial q_x}dfrac{partial }{partial p_x}+dfrac{partial p_y}{partial q_x}dfrac{partial }{partial p_y}, , ,dfrac{partial p_x}{partial q_y}dfrac{partial }{partial p_x}+dfrac{partial p_y}{partial q_y}dfrac{partial }{partial p_y}proper)
finish{align*}
$$
g_{q(p)}=left(left(Dq(p)proper)^{-1}proper)^tau g(p)left(Dq(p)proper)^{-1}
$$
The tangent vectors in our instance are already orthogonal and each diagonal entries are in every single place constructive. ##g_p## has due to this fact the signature ##(+,+)## and is a Riemannian metric.

What’s a size based on a metric tensor? The road component will be regarded as a line section related to an infinitesimal displacement vector
$$
ds^2=sum_{i,j}g_{ij}dx_idx_j=g_{ij}dx^idx^j=g_{ij}dfrac{dx^i}{dt}dfrac{dx^j}{dt}dt^2 =g[mathbf{f}]
$$
because it defines the arclength by
$$
s=int_a^b sqrtds^2=int_a^bsqrt{left|g_{ij}(gamma(t))left(dfrac{d}{dt}x^i circ gamma(t)proper)left(dfrac{d}{dt}x^j circ gamma(t)proper)proper|},dt.
$$
The road component utterly defines the metric tensor since we thought-about arbitrary curves. That’s the reason it’s typically used as a characterization of the metric tensor somewhat than the precise tensor product. In our instance we’ve got (with ##X_1=D_p(x)=dp_x## and ##X_2=D_p(y)=dp_y##)
$$
ds^2=g=(cosh^2 p_x +sinh^2 p_y ),dp_x^2+(1+4p_x^2+4p_y^2)^{-1},dp_y^2
$$
The notation of the coordinate features as ##p_x,p_y## are supposed to exhibit that we’re contemplating actual valued features ##Mto mathbb{R}.## The standard notation is
$$
ds^2=g=(cosh^2 x +sinh^2 y ),dx^2+(1+4x^2+4y^2)^{-1},dy^2.
$$
The metric tensor at ##(0,0,0)## is given as ##g(0,0,0)=ds^2=dx^2+dy^2## and due to this fact the Euclidean metric. The angle between the 2 foundation vectors is ##cossphericalangleleft( X_1(0,0,0),X_2(0,0,0)proper)=0,## i.e. the premise vectors are orthogonal. At ##p=(2,1,3)## we’ve got a distinct metric however nonetheless an orthogonal foundation since ##g(2,1,3)((1,0),(0,1))=0## as a result of the truth that ##g[mathbf{f}]## is diagonal and ##[mathbf{f}]## an orthogonal body. However the lengths of the premise vectors are now not ##1.##
start{align*}
X_1(2,1,3)&=|(1,0)|_{(2,1,3)}=sqrt{cosh^2(2)+sinh^2(1)} approx 3.94
X_2(2,1,3)&=|(0,1)|_{(2,1,3)}= sqrt{1/21}approx 0.22
finish{align*}

Differential Geometry

The differential geometer and the physicist share the toolbox, however not essentially the angle. The primary distinction is likely to be the diploma of abstraction. Emmy Noether spoke of impartial variables ##p_x,p_y##, dependent features ##q_x(p_x,p_y),q_y(p_x,p_y),## and the appliance of a bunch of transformations. Trendy differential geometers communicate of sections and principal fiber bundles, or Euler-Lagrange equations and differential operators. Emmy Noether’s unique paper [9] is to some lengthen simpler to know than the fashionable therapy of her well-known theorem, cp. [11].

The graph of the perform ##f:Blongrightarrow {z}## is ##{(x,y,x^2-y^2)}=B instances {z}=Econg mathbb{R}^3.## ##B## known as the bottom house, right here the ##(x,y)##-plane, and ##E## the full house, right here the whole ##mathbb{R}^3.## We now have a pure projection ##pi:Elongrightarrow B, , ,(x,y,z)mapsto (x,y).## Any clean perform ##sigma :Blongrightarrow E## such that ##pi(sigma (x,y))=(x,y)## known as a piece, and the set of all sections is denoted by ##Gamma(E).## However what does this must do with differentiation and the saddle? We already launched the (tangent) vector subject, which can also be known as tangent bundle or simply vector bundle.

$$
E=TM=bigsqcup_{pin M}T_pM=bigcup_{pin M}{p}instances T_pM
$$
the place the tangent areas are the bottom house isomorphic to ##mathbb{R}^2## and ##sigma(p_x,p_y)=(p_x,p_y,p_x^2-p_y^2)## are the sections. The preimage ##pi^{-1}(p_x,p_y)={p}instances T_pM## known as fiber (over ##p=(p_x,p_y)##). Bundle as a result of we take into account units of factors, i.e. bundles (units) of fibers. That is additionally the rationale why we use the notation with ##sqcup_p.## To be exact, we should always have launched the cotangent bundle as a substitute as a result of our metric tensor is outlined as
$$
Gamma((TMotimes TM)^*ni g , , ,g_p in T_p^*Motimes T_p^*M = (T_pMotimes T_pM)^*
$$
If we’ve got a bunch working on the fibers then we communicate of principal bundles. Emmy Noether’s group of clean transformations is the group that operates on a precept bundle, and its smoothness qualifies it as a Lie group.

Levi-Civita Connection

A connection is a directional spinoff by way of vector fields comparable with the gradient. The Levi-Civita connection is torsion-free and preserves the Riemannian metric tensor. It’s also known as the covariant spinoff alongside a curve
$$
nabla_{dotgamma(0)} sigma(p) =(sigma circ gamma )'(0)=lim_{t to 0}dfrac{sigma (gamma (t))-sigma (p)}{t},
$$
or Riemannian connection. Its existence and uniqueness are generally known as the basic theorem of Riemmanian geometry. An (affine) connection is formally outlined as a bilinear map
$$
nabla, : ,Gamma(TM)timesGamma(TM) longrightarrow Gamma(TM), , ,(X,Y)longmapsto nabla_XY
$$
that’s ##C^infty (M)##-linear within the first argument, ##mathbb{R}##-linear within the second, and obeys the Leibniz rule ##nabla_X(hsigma )=X(h)cdot sigma +hcdot nabla_X(sigma).## The Levi-Civita connection is an affine connection that’s torsion-free, i.e. ##nabla_XY-nabla_YX=[X,Y]## with the Lie bracket of vector fields, and preserves the metric, i.e.
$$
Z(g(X,Y))=g(nabla_Z X,Y)+g(X,nabla_Z Y))textual content{ or quick }nabla_Z g(X,Y)=0.
$$
The distinction to a Lie spinoff is, {that a} Lie spinoff is ##mathbb{R}##-linear however doesn’t must be ##C^infty (M)##-linear. We additionally point out the Koszul method
start{align*}
g(nabla_{X}Y,Z)&={tfrac{1}{2}}{Large {}X{bigl (}g(Y,Z){bigr )}+Y{bigl (}g(Z,X){bigr )}-Z{bigl (}g(X,Y){bigr )}&phantom{=}+g([X,Y],Z)-g([Y,Z],X)-g([X,Z],Y){Large }}.
finish{align*}
If we write with Christoffel symbols ##Gamma_{ij}^ok##
$$
nabla_jpartial_k =Gamma_{jk}^lpartial_l
$$
then $$
nabla_XY=X^jleft(partial_j(Y^l)+Y^kGamma_{jk}^lpartial_lright) $$
and ##nabla## is appropriate with the metric if and provided that
$$partial_{i}{bigl (}g(partial_{j},partial_{ok}){bigr )}=g(nabla_{i}partial_{j},partial_{ok})+g(partial_{j},nabla_{i}partial_{ok})=g(Gamma_{ij}^{l}partial_{l},partial_{ok})+g(partial_{j},Gamma_{ik}^{l}partial_{l}),$$

i.e. ##partial_{i}g_{jk}=Gamma_{ij}^{l}g_{lk}+Gamma_{ik}^{l}g_{jl}.## It’s torsion-free if

$$
nabla_{i}partial_{j}-nabla_{j}partial_{i}=(Gamma_{jk}^{l}-Gamma_{kj}^{l})partial_{l}=[partial_{i},partial_{j}]=0,$$

i.e. ##Gamma_{jk}^{l}=Gamma_{kj}^{l}.## We are able to examine by direct calculation that
$$Gamma_{jk}^{l}={tfrac{1}{2}}g^{lr}left(partial_{ok}g_{rj}+partial_{j}g_{rk}-partial_{r}g_{jk}proper)$$

the place ##g^{lr}## stands for the inverse metric ##(g_{lr})^{-1}.##

Let ##gamma(t):[-1,1]longrightarrow M## be a clean curve on the manifold ##M.## A bit ##sigma in Gamma(TM)## alongside ##gamma ## known as parallel based on ##nabla## if
$$nabla_{dotgamma(t)}sigma (gamma(t))=0 textual content{ for all } t.$$
This implies in case of tangent vectors of tangent bundles of a manifold that each one tangent vectors are fixed with respect to an infinitesimal displacement from ##gamma(t)## within the course of ##dotgamma(t).## A vector subject known as parallel whether it is parallel with respect to each curve within the manifold. Let ##t_0,t_1in [-1,1].## Then there’s a distinctive parallel vector subject ##Y:Mrightarrow T_{gamma(t)}M## alongside ##gamma ## for each ##X_pin T_{gamma(t_0)}M## such that ##X_p=Y_{gamma (t_0)}.## The perform
$$ P_{gamma(t_{0}),gamma(t_{1})}colon T_{gamma(t_{0})}Mto T_{gamma(t_{1})}M,quad X_pmapsto Y_{gamma(t_{1})} $$
known as parallel transport. The existence and uniqueness comply with from the worldwide model of the Picard-Lindelöf theorem for unusual linear differential equation methods. We now have completely different internal merchandise at ##t_0## and ##t_1,## and thus completely different metrics. A parallel transport determines what occurs alongside a path from one tangent house to the opposite.

We now have commuting two-dimensional vector fields in our instance of the two-dimensional saddle ##M## as a result of the partial derivatives commute. This implies we’ve got ##nabla_{X_1}X_2=nabla_{X_2}X_1.## Moreover, the metric is diagonal, i.e. solely stretches and compressions alongside the coordinate features. Let
$$
%gamma: [-1,1] rightarrow M, , ,gamma(t)=(2t+1,t+1/2,3(t+1/2)^2)
gamma : [-1,1] rightarrow M, , ,gamma(t)=(2t+1,t+1/2)
$$
with ##p=gamma(-1/2)=(0,0)## and ##q=gamma(1/2)=(2,1).## We have to resolve
$$
nabla_{(2,1)}sigma(2t+1,t+1/2)= lim_{t’ to t}dfrac{sigma(2t’+1,t’+1/2)-sigma(2t+1,t+1/2)}{t’} =0
$$
which is clearly true for a (clean) part of ##Gamma(TM).## The parallel transport is solely given by holding the elements of the vector fields. ##X_p=alpha X_1(p)+beta X_2(p)## maps to ##Y_{q}=alpha X_1(q)+beta X_2(q).## This implies within the three dimensions of the embedding
$$
start{pmatrix}alpha beta finish{pmatrix}longmapsto start{pmatrix}2t+1t+1/23(t+1/2)^2end{pmatrix}+start{pmatrix}alpha beta 4alpha t +2alpha -2beta t-beta finish{pmatrix}.
$$