On the Stochastic Flows on $(m+n+1)$-Dimensional Exotic Spheres
Nurfarisha Adhitya Ronnie Effendie Muhammad Farchani Rosyid
math-ph math.AP math.AT math.DG math.MP math.PR 60H10, 57S15, 51H25

Exponential suppression of bit-flips in a qubit encoded in an oscillator
    Raphaël Lescanne^1,2, Marius Villiers^1,2, Théau Peronnin^3, Alain Sarlette^2, Matthieu Delbecq^1, Benjamin Huard^3, Takis Kontos^1, Mazyar Mirrahimi^2, Zaki Leghtas^4, 1, 2
    December 30, 2023
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Stochastic flows of Stratonovich stochastic differential equations on exotic spheres have been studied. The consequences of the choice of exotic differential structure on stochastic processes taking place on the topological space S^m+n+1 as state space of the processes have been investigated. More precisely, we have investigated the properties of stochastic processes where the state spaces of the stochastic processes under consideration are (m+n+1)-dimensional differentiable manifolds which are homeomorphic but not necessarily diffeomorphic to standard (m+n+1)-dimensional sphere. The differentiable manifolds have been constructed from disjoint union ℝ^m+1× S^n⊔ S^m×ℝ^n+1 by identifying every pair of its points using a map u :ℝ^m+1× S^n→ S^m×ℝ^n+1 which is constructed from a diffeomorphism h_1× h_2:S^m× S^n→ S^m× S^n. The diffeomorphisms h_1 and h_2, therefore, can be regarded as the carriers of the "exoticism" of the constructed manifolds. For all of the above purposes, homeomorphisms h from the above-constructed manifolds onto the standard sphere explicitly in term of the diffeomorphisms h_1 and h_2 have been constructed. Using the homeomorphisms h and all their associated maps derived from them and expressed in terms of h_1 and h_2 as well as their derivatives, we construct the stochastic processes or flows on the above-constructed manifolds corresponding to stochastics processes on the standard sphere S^m+n+1_s. The stochastic processes yielded from the above construction on the constructed manifolds can be regarded as the same stochastic processes on S^m+n+1_s but described in exotic differential structures on S^m+n+1.

Keywords: Stochastic flow; stratonovich stochastic differential equation; exotic spheres; exotic differential structure; homeomorphism; Sobolev regularities.

Mathematics Subject Classification (2010): 60H10; 57S15; 51H25.









§ INTRODUCTION

Stochastic processes appear in many physical phenomena, such as the well-known diffusion process, heat conduction, Brownian motion, the formation of river meander, the fluctuation current due to thermal noise, etc. Mathematically, a stochastic process is a set of random variables parametrized by another set (discrete or continuous) playing the role of time. A stochastic process is a solution of so-called stochastic differential equations. Since there exists a distribution function for every random variable, there is an associated distribution function depending on the time for every stochastic process. The distribution function associated with a stochastic process satisfies a deterministic partial differential equation called the Fokker-Planck equation associated with the process which is obtained from the stochastic differential equation governing the process via Itô equation. A Fokker-Planck equation is used by Einstein to define the time evolution of the probability density of particles whereas Langevin rewrites an explicit deterministic momentum equation for a particle augmented by a Gaussian white noise force which perturbs the particle trajectory. The approach of Langevin now constitutes a canonical stochastic differential equation for a full scope of systems, but since its introduction, the mathematical and physical interpretation of the white noise term has been discussed and debated. The white noise interpretation of stochastic differential equations naturally leads to stochastic differential equations in Stratonovich sense. This is because discrete-time and smooth approximations of white noise drive stochastic differential equations to converge to stochastic differential equations in Stratonovich sense, not in Itô sense.


The microscopic description of the dynamics of a diffusion process on a connected compact differentiable manifold M, for instance, is typically represented by a so-called Stratonovich formulation of stochastic differential equations <cit.>. Likewise, the dynamic descriptions of the other stochastic processes on differentiable manifolds are also represented in the formulation. The advantage of this formulation is that Itô's formula appears in the same form as the fundamental theorem of calculus; therefore, stochastic calculus in this formulation takes a more familiar form. The other advantage of Stratonovich formulation, which is more important and relevant to our investigation is that stochastic differential equations on manifolds in this formulation transform consistently under diffeomorphisms between manifolds <cit.>. Here we will consider a Stratonovich's stochastic differential equation on compact Riemannian manifold. There are at least two ways to solve the differential equation. Firstly, we solve it locally (i.e. all mathematical terms appearing in the equation are represented locally concerning coordinate systems covering the manifold, and these local representations of the equation are solved simultaneously) and then patch them up. In the second one, we obtain the solution of the equation in a suitable embedding space ℝ^N and prove that the solution stays in M whenever the starting point q is a point in M. Both methods work well only in the case that the vector fields appearing in the above differential equation as coefficients are smooth (at least C^2-differentiable). In the case of non smooth drifting vector field, i.e. it has only Sobolev regularity and bounded divergence, Zhang <cit.> has point out the existence and uniqueness of a so-called invertible ν-almost everywhere stochastic invertible flows of the stochastic differential equation, where ν is the Riemannian measure on the manifold under study. The starting point of Zhang's proof is the constructions of a sequence of smooth vector fields, which is convergent in Sobolev norm to the drifting vector field. The construction of such a sequence is possible because the drifting vector field is assumed to be a Sobolev vector field. The sequence of smooth vector fields leads to a sequence of stochastic differential equations which "converges" to the initial stochastic differential equation. Every stochastic differential equations in the sequence have a unique solution. Assuming the boundedness of the drifting vector field and its divergence, the sequence of the solutions of the stochastic differential equations in the sequence converges in a suitable metric constructed from geodesic metric to the unique invertible ν-almost everywhere stochastic invertible flows.


In general, the laws of physics must be independent of the choice of a coordinate system. This statement means, for instance, that there must be a family of coordinate systems that are compatible with describing space-time, i.e. the space of all events. A set of compatible coordinate systems in space-time is a subset of a so-called differential structure. Space-time is an example of a topological space equipped with a differential structure <cit.>. In a topological space, we may find more than one differential structures. Therefore, from a single topological space of all events, we can construct more than one space-time with different differential structures. If the yielded space-times are not distinguishable in the sense that they are not diffeomorphic, they may lead to inequivalent formulations of the law of physics. The totality of inequivalent differential structures on a topological space is called the exotica of differential structures on the topological space, and a differential structure that is not equivalent to the standard one is called an exotic differential structure on the topological space under consideration <cit.>. Unfortunately, among the exotic differential structures whose existences are already known on various topological spaces, only the exotic differential structures on spheres can be constructed explicitly. Therefore, the exotic spheres can be used as toy models serving as configuration or phase spaces of physical systems as well as the base manifolds for various field theories. Various problems of real interest can be answered by studying the models and the results they are compared to those obtained from the standard spheres <cit.>.



In mathematics, spheres are topological spaces which are interesting to investigate. They serve, for instance, as models for configuration spaces of some mechanical systems. In physics, for example, the standard 7-dimensional sphere S^7_s is particularly interesting in related to supersymmetry breaking <cit.> and to the work of Witten <cit.> in which he used it to cancel the global gravitational anomalies in 1985. The standard seven-dimensional sphere is also regarded as the total space of a principal SU(2) bundle of Yang-Mills theory <cit.>. Two differential structures on  7-dimensional sphere S^7 are said to be equivalent if there is a diffeomorphism pulling back the differentiable maximal atlas from the second to the first. The connected compact topological space S^7 has more than one distinct differential structure that are not equivalent each other in this sense or more precisely there are connected compact 7-dimensional differentiable manifolds which are homeomorphic but not diffeomorphic to the standard seven-sphere S^7_s <cit.>. The differential structure on the standard sphere S^7_s is called a standard differential structure, while the differential structures that are not equivalent to the standard one are called exotic differential structures. The topological space S^7 equipped with an exotic differential structure or a seven-dimensional differentiable manifold which is homeomorphic but not diffeomorphic to standard seven-dimensional sphere S^7_s is called exotic 7-sphere. 


The main purpose of this work is to understand the consequences of the choice of exotic differential structure on stochastic processes taking place on the topological space S^m+n+1 as state space of the prosesses. More precisely, we investigate the properties of stochastic processes where the state spaces of the stochastic processes under consideration are (m+n+1)-dimensional differentiable manifolds M^m+n+1_(h_1,h_2) which are homeomorphic but not necessarily diffeomorphic to standard (m+n+1)-dimensional sphere S^m+n+1_s. A manifolds M^m+n+1_(h_1,h_2) is constructed (see <cit.>) from disjoint union ℝ^m+1× S^n⊔ S^m×ℝ^n+1 by identifying every pair of its points using a map u :ℝ^m+1× S^n→ S^m×ℝ^n+1 which is constructed from a diffeomorphism h_1× h_2:S^m× S^n→ S^m× S^n. Whether a manifold M^m+n+1_(h_1,h_2) is homeomorphic or even diffeomorphic to the standard sphere S^m+n+1_s depends on the diffeomorphism h_1, :S^m→ S^m and  h_2:S^n→ S^n. We consider here only the cases where the diffeomorphisms h_1 and h_2 lead to the manifolds M^m+n+1_(h_1,h_2) which are at least homeomorphic to standard sphere. The diffeomorphisms h_1 and h_2 therefore can be regarded as the carriers of the "exoticism" of M^m+n+1_(h_1,h_2). For all of the above purposes, we firstly construct from the diffeomorphisms h_1 and h_2 a homeomorphism h:M^m+n+1_(h_1,h_2)→ S^m+n+1_s explicitly in term of the diffeomorphisms h_1 and h_2. The homeomorphism h topologically identifies the topological space M^m+n+1_(h_1,h_2) with S^m+n+1. For instance, if the diffeomorphisms h_1 and h_2 determining M^m+n+1_(h_1,h_2) are the identity maps, then the corresponding homeomorphism h is a diffeomorphism. Assuming a stochastic differential equation which satisfies certain regularities on the standard spheres, then associated to the equation there exists a unique stochastic flow on the standard sphere in the sense of Zhang <cit.>. Using the identification h, a stochastic flow can be constructed on M^m+n+1_(h_1,h_2). The stochastic flow yielded from the above construction on M^m+n+1_(h_1,h_2) can be regarded as the same stochastic flow on S^m+n+1_s but described in exotic differential structures on S^m+n+1. In turn, by making use of the homeomorphism h and associated maps derived from it (expressed in terms of h_1 and h_2 as well as their derivatives), we construct a stochastic differential equation on M^m+n+1_(h_1,h_2) corresponding to the stochastics differential equation on the standard sphere S^m+n+1_s.  We investigate then the stochastic flow yielded in the above mentioned construction on the exotic spheres. The main result of our investigation is depicted than in Theorem <ref>.




§ MILNOR'S CONSTRUCTION OF EXOTIC SPHERES AND THEIR HOMEOMORPHISM INTO STANDARD SPHERE


Let h_1× h_2:S^m× S^n→ S^m× S^n be a diffeomorphisme from S^m× S^n onto itself so that every (x,y)∈ S^m× S^n is maped into (h_1(x), h_2(y))∈ S^m× S^n. Furthermore, define a map u :ℝ^m+1× S^n→ S^m×ℝ^n+1 defined by u(tx,y)=(h_1(x),t^-1h_2(y)) for every t∈ (0,∞ ) and (x,y)∈ S^m× S^n. The manifold M^m+n+1_(h_1,h_2) is obtained from ℝ^m+1× S^n⊔ S^m×ℝ^n+1 by gluing or matching every (tx,y)∈ℝ^m+1× S^n with its image u(tx,y)=(h_1(x),t^-1h_2(y)) under the map u. It is an (m+n+1)-dimensional manifold. Wether the obtained manifold M^m+n+1_(h_1,h_2) is homeomorphic or even diffeomorphic to S^m+n+1_s depends merely on the maps h_1 and h_2.



Now consider two maps defined by

    ℝ^m+1× S^n∋ (x̃,ỹ)↦(x̃/√(1+x̃^2), ỹ/√(1+x̃^2))∈ S^m+n+1,

and

    S^m×ℝ^n+1∋ (x̅,y̅)↦(h_1^-1(x̅)/√(1+y̅^2), y̅h_2^-1(y̅/y̅)/√(1+y̅^2))∈ S^m+n+1.

If 

    x̅=h_1(x̃/x̃)   andy̅=h_2(ỹ)/x̃,

i.e. (x̃,ỹ) and (x̅,y̅) represent the same point in M^m+n+1_(h_1,h_2), then it is clear that y̅=x̃^-1 and

    (x̃/√(1+x̃^2), ỹ/√(1+x̃^2))=
    (h_1^-1(x̅)/√(1+y̅^2), y̅h_2^-1(y̅/y̅)/√(1+y̅^2)).

Therefore the maps h:M^m+n+1_(h_1,h_2)→ S^m+n+1_s defined by Eq.(<ref>) and Eq. (<ref>) is well defined.

Now let (x̃,ỹ) and (x̆,y̆) be any two elements of ∈ℝ^m+1× S^n with (x̃,ỹ) ≠ (x̆,y̆). If 


    (x̃/√(1+x̃^2), ỹ/√(1+x̃^2)) = (x̆/√(1+x̆^2), ỹ/√(1+x̆^2)),

then 

    x̃/√(1+x̃^2) =  x̃/√(1+x̆^2)andỹ/√(1+x̃^2) =  ỹ/√(1+x̆^2).

Since ỹ and y̆ are elements of S^n, we have

    √(1+x̃^2) =  √(1+x̆^2).

From Eq.(<ref>) and Eq.(<ref>) we have (x̃,ỹ)= (x̆,y̆). It contradicts the above asummption. Therefore, the map h is an injection.

Let (γ, κ)∈ S^m+n+1_s. Furthermore let (x,y)∈ S^m× S^n and t∈ [0,∞ ) such that

    γ=tx/√(1+t^2)andκ =y/√(1+t^2).

From Eq. (<ref>) we have


    γ^2(1+t^2)=t^2x^2=t^2,

and

    κ^2(1+t^2)=y^2=1,

since x∈ S^m and y∈ S^n. From Eq.(<ref>) and Eq.(<ref>) we obtain then

    t=√(γ^2/κ^2)=γ/κ,


    tx   =   γ/κ  γ√(κ^2/γ^2+1)=γ√(1+γ^2/κ^2),

and

    y   =   κ√(1+γ^2/κ^2).

This means the map h is surjective and therefore it is a bijection.
Let f denote the invers of h. Thus, f:S^m+n+1→ M^m+n+1_(h_1,h_2) is given by

    S^m+n+1∋ (γ, κ)   ↦   (γ/κ  γ√(κ^2/γ^2+1) , κ√(1+γ^2/κ^2))∈ℝ^m+1× S^n
    
    
    (h_1[γ√(κ^2/γ^2+1) ], κ/γ h_2[κ√(1+γ^2/κ^2) ])∈ S^m×ℝ^n+1.






§ STOCHASTIC FLOWS ON M^M+N+1_(H_1,H_2)

Let (W_t)_t≥ 0 denote the d-dimensional standard Brownian motion on the classical Wiener space (Ω , ℱ,P; (ℱ_t)_t≥ 0), where Ω is the space of all continuous function from ℝ_+ to ℝ^d with locally uniform convergence topology, ℱ is the Borel σ-field on Ω generated by the topology, P is the Wiener measure on ℱ, and (ℱ_t)_t≥ 0 is the filtration generated by W_t(ω )=ω _t. Now consider the Stratonovich's stochastic differential equation 

    dq_t=X_0(q_t)dt+∑ _k=1^dX_k(q_t)∘ dW^k_t,q_0=q

on a compact differentiable Riemannian manifold (M, g), where X_i, i=0,... d are d+1 vector fields on M. There are at least two ways to solve the differential equation. In the first way, we solve it locally and then patch them up. In the second one, we obtain the solution of the equation in a suitable embeding space ℝ^N and prove that the solution stays in M whenever the starting point q∈ M. It is well-known that the two ways as mentioned above work well only if the drift vector field X_0 and the other vectors X_k (k=1,...d) appearing in the above differential equation are smooth (at least C^2). For the case of X_0 having only Sobolev regularity and bounded divergence and X_k (k=1,⋯,d) are C^2, Zhang <cit.> has shown the existence and uniqueness of the so-called ν-almost everywhere stochastic invertible flows of SDE(<ref>), where ν is the Riemannian measure on M.



(Zhang <cit.>)
	Let q_t(ω ,q) be an M-valued measurable stochastic field on ℝ_+×Ω× M. The flow q_t(q) is called a ν-almost everywhere stochastic flow of (<ref>) corresponding to vector fields X_i (i=0,...d) if
	
		
  * For ν-almost all q∈ M, t↦ q_t(q) is continuous and (ℱ_t)-adapted stochastic process and, satisfies that for any t>0 and ζ∈ C^∞ (M)
		
    ζ(q_t(q))=ζ (q)+∫ _0^tX_0ζ (q_s(q))ds+∫ _0^tX_kζ (q_s(q))∘ dW^k_s∀ t≥ 0.

		
  * For arbritary t≥ 0 and P-almost all ω∈Ω, (ν∘ q_t)(ω ,·)≪ν. For any T>0. there is a positive constant K_T,X_0,X_k such that for all non-negative measurable function ζ on M
		
    sup _t∈ [0,T]𝔼∫ _Mζ (q_t(q))ν (dq) ≤ K_T,X_0,X_k∫ _M ζ (q)ν (dq).

	
	Furthermore, the ν-almost everywhere stochastic flow q_t(q) of (<ref>) is said to be invertible if for all t≥ 0 and P-almost all ω∈Ω, there exists a measurable invers q_t^-1(ω , · ) of  q_t(ω ,· ) so that ν∘ q_t^-1(ω , · )=ρ _t(ω ,· )ν, where the density ρ _t(q) is given by
	
    ρ _t(q)=exp[∫ _0^t div X_0(q_s(q))ds + ∫ _0^t div X_k(q_s(q))∘ dW^k_s ].




Let C^k(TM) be the set of all C^k-differentiable vector fields on M, for every k∈ℕ∪{∞}. For every p≥ 1, we define

    X_p:=[∫ _M|X(q)|^pν (dq)]^1/p,

and

    X_1,p:= X_p+[∫ _M|∇ X(q)|^pν (dq)]^1/p,

for every X∈ C^∞ (TM), where ∇ is the Levi-Civita connection associated to the metric tensor g on M. The completions of C^∞ (TM) with respect to ·_p and ·_1,p will be denoted by L^p(TM) and ℍ^p_1(TM) respectively. Then Zhang found the following fact.
(Zhang <cit.>)
	Let X_0∈ L^∞ (TM)∩ℍ^p_1(TM) for some p≥ 1, div  X_0∈ L^∞ (M), and for each k=1,⋯ ,d, X_k∈ C^2(TM). Then there exists a unique ν-almost everywhere stochastic invertible flow q_t(q) of (<ref>) in the sense of Definition <ref>



Now consider the case where M is the standard sphere S^m+n+1_s provided by the natural Riemannian metric tensor g induced from Euclidean metric in ℝ^m+n+2. Let V_0∈ L^∞ (TS^m+n+1_s)∩ℍ^p_1(TS^m+n+1_1) and V_k (k=1,⋯ , d) be C^2-vector fields on S^m+n+1. Then according to Theorem <ref>, there exists a unique ν-almost everywhere stochastic invertible flow q_t(q) of the following stochastic differential equation

    dq_t=V_0(q_t)dt+∑ _k=1^dV_k(q_t)∘ dW^k_t,q_0=q

on S^m+n+1_s. With the identivication h and its invers f constructed in the previous section, then (f∘ q_t∘ h)(q̅) is a stochastic flow on M^m+n+1_h_1,h_2, where q̅ is the element of M^m+n+1_h_1,h_2 which is identified with q, i.e., h(q̅)=q. The stochastic flow (f∘ q_t∘ h)(q̅) and the stochastic flow q_t(q) is actually the same flow on the topological space S^m+n+1. The flow q_t(q) is the stochastic flow on S^m+n+1 described using the standar differential structure on S^m+n+1. Whereas, the flow (f∘ q_t∘ h)(q̅) can be regarded as the same stochastic flow on S^m+n+1 described using another differential structure which may be exotic (depending on the maps h_1 and h_2). If the maps h_1 and h_2 leads to manifold M^m+n+1_h_1,h_2 which is diffeomorphic to S^m+n+1_s so that h is a diffeomorphism, then (f∘ q_t∘ h)(q̅) and q_t(q) clearly are the same ν-almost everywhere stochastic invertible flow on S^m+n+1_s. Now we will investigate the properties of the flow (f∘ q_t∘ h)(q̅) for the general case in which M^m+n+1_h_1,h_2 may not be diffeomorphic but homeomorphic to S^m+n+1_s. 


The vector fields V_0,V_1, ⋯ , V_d on S^m+n+1_s appearing in equation (<ref>) are vector fields on S^m+n+1 when they are described using the standard differential structure on S^m+n+1. The vector fields are seen as f_∗ V_0, f_∗ V_1, ⋯, f_∗ V_d on M^m+n+1_h_1,h_2 if they are described by using another (exotic) differential structure on S^m+n+1. The stochastic differential equation (<ref>) is seen as

    d(f∘ q_t∘ h)=f_∗ V_0(f∘ q_t∘ h)dt+∑ _k=1^d f_∗ V_k(f∘ q_t∘ h)∘ dW^k_t,f∘ q_0∘ h=q̅

on M^m+n+1_h_1,h_2. The properties of the flow f∘ q_t∘ h can be understood from the equation (<ref>), especially from the vector fields appearing in the equation.
At point (x̅,y̅)∈ M^m+n+1_h_1,h_2 the value of vector field f_∗ V_k is given by

    (f_∗ V_k)(x̅, y̅)=(f_∗)_(γ ,κ )[V_k(γ ,κ )],

where (γ ,κ )=h(x̅,y̅). If V_k^μ (μ=1,2, ⋯ , m+n+2) is the components of V_k in the ambient space ℝ^m+n+1, then for ν =1,2,⋯ ,m+n+2

    (f_∗)_(γ ,κ )[V_k(γ ,κ )]^ν = ∑ _j=1^m+1∂ f^ν/∂γ ^jV_k^j(γ ,κ )+∑ _r=1^n+1∂ f^ν/∂κ ^rV_k^r(γ ,κ ).

If (x̅,y̅) is in the region S^m×ℝ^n+1 of M^m+n+1_h_1,h_2, the differentials of f are given by

    ∂ f^ν/∂γ^j    =    ∑_i=1^m+1∂ h_1^ν(γ̂)/∂γ̂^i (δ_j^i/γ-κ^2/γ^3 γ^iγ^j)
       =   ∂ h_1^ν/∂γ̂^j√(1+y̅^2)-∑ _i=1^m+1∂ h_1^ν/∂γ̂^iy̅^2/√(1+y̅^2)[h_1^-1(x̅)]^i[h_1^-1(x̅)]^j     (ν, j=1,..., m+1),
    ∂ f^ν/∂κ^r    =    ∑_i=1^m+1∂ h_1^ν(γ̂)/∂γ̂^i γ^iκ^r/γ
       =   ∑ _i=1^m+1∂ h_1^ν/∂γ̂^iy̅/√(1+y̅^2)[h_1^-1(x̅)]^i[h_2^-1(y̅/y̅)]^r     (ν=1,..., m+1,  r=1,...,n+1),
    ∂ f^ν/∂γ^j   =   -κ/γ^3h_2^ν-m-1((κ̂)γ^j+∑_s=1^n+1∂ h_2^ν-m-1(κ̂)/∂κ̂^sκ^sγ^j/γ
       =   -y̅^ν-m-1√(1+y̅^2)[h_1^-1(x̅)]^j+
    ∑_s=1^n+1∂ h_2^ν-m-1/∂κ̂^s[h_1^-1(x̅)]^j[h_2^-1(y̅/y̅)]^s/√(1+y̅^2),

(j=1,...,m+1, ν=m+2,..., m+n+2), and

    ∂ f^ν/∂κ^r   =   h_2^ν-m-1(κ̂)κ^r/γκ+∑_s=1^n+1∂ h_2^ν-m-1(κ̂)/∂κ̂^s(δ^s_r/γ-γ/κ^2κ^sκ^r)
       =   y̅^ν-m-1√(1+y̅^2)[h_2^-1(y̅/y̅)]^r 
           +∑ _s=1^n+1∂ h_2^ν-m-1/∂κ̂^s(δ ^s_r√(1+y̅^2)-[h_2^-1(y̅/y̅)]^s[h_2^-1(y̅/y̅)]^r/√(1+y̅^2)),

(r=1,...,n+1, ν =m+2,...,m+n+2), since 
    h_2^ν-m-1(κ̂)=h_2^ν-m-1[h_2^-1(y̅/y̅)]=y̅^ν-m-1/y̅.

If (x̅,y̅) is in the region ℝ^m+1× S^n of M^m+n+1_h_1,h_2, the differentials of f are given by

    ∂ f^ν/∂γ ^j   =   γ ^νγ ^j/κ+δ ^ν_j/κ,
       =   x̅^νx̅^j/√(1+x̅^2)+δ ^ν _j √(1+x̅^2)     (ν ,j=1,..., m+1),
    ∂ f^ν/∂κ ^r   =   -γ^2γ ^νκ  ^r/κ^3, 
       =   -x̅^2x̅^νy̅^r/√(1+x̅^2)     (ν =1,...,m+1,  r=1,...,n+1),
    ∂ f^ν/∂γ ^j   =   κ ^ν-m-1γ ^j/κ,
       =   x̅^jy̅^ν-m-1/√(1+x̅^2) (ν =m+2,...,m+n+2,  j=1,..., m+1),
    ∂ f^ν/∂κ ^r   =   δ ^ν-m-1 _r/κ-γ^2κ ^ν-m-1κ ^r/κ^3
       =   √(1+x̅^2)δ _r^ν -m-1-x̅^2y̅^ry̅^ν -m-1/√(1+x̅^2),

(ν =m+2,...,m+n+2,  r=1,...,n+1).
In the above expressions, γ̂:=γγ^-1=(γ ^1,...,γ ^m+1)γ^-1 and κ̂:=κκ^-1=(κ ^1,...,κ ^n+1)κ^-1. Then, the vector fields (f_∗ V_k) (k=0,1,...,d) at (x̅, y̅)∈ S^m×ℝ^n+1 are given by

    [(f_∗ V_k)(x̅, y̅)]^ν   =   ∑ _j=1^m+1[∂ h_1^ν/∂γ̂^j√(1+y̅^2)-∑ _i=1^m+1∂ h_1^ν/∂γ̂^iy̅^2/√(1+y̅^2)[h_1^-1(x̅)]^i[h_1^-1(x̅)]^j]V_k^j(h(x̅,y̅))
           +∑ _r=1^n+1[∑ _i=1^m+1∂ h_1^ν/∂γ̂^iy̅/√(1+y̅^2)[h_1^-1(x̅)]^i[h_2^-1(y̅/y̅)]^r]V_k^r(h(x̅,y̅)),

for ν =1, ..., m+1, and

    [(f_∗ V_k)(x̅, y̅)]^ν   =   ∑ _j=1^m+1[-y̅^ν-m-1√(1+y̅^2)[h_1^-1(x̅)]^j]V_k^j(h(x̅,y̅)) 
           + ∑ _j=1^m+1[∑_s=1^n+1∂ h_2^ν-m-1/∂κ̂^s[h_1^-1(x̅)]^j[h_2^-1(y̅/y̅)]^s/√(1+y̅^2)] V_k^j(h(x̅,y̅))
          +∑ _r=1^n+1[y̅^ν-m-1√(1+y̅^2)[h_2^-1(y̅/y̅)]^r]V_k^r(h(x̅,y̅))
          +∑ _r=1^n+1[∑ _s=1^n+1∂ h_2^ν-m-1/∂κ̂^s(δ ^s_r√(1+y̅^2)-[h_2^-1(y̅/y̅)]^s[h_2^-1(y̅/y̅)]^r/√(1+y̅^2))]V_k^r(h(x̅,y̅)),

for ν=m+2, ..., m+n+2. Whereas, at (x̅, y̅)∈ℝ^m+1× S^n the value of vector fields (f_∗ V_k) (k=0,1,...,d) are given by

    [(f_∗ V_k)(x̅, y̅)]^ν    =    ∑ _j=1^m+1[x̅^νx̅^j/√(1+x̅^2)+δ ^ν _j √(1+x̅^2)]V_k^j(h(x̅,y̅))
          + ∑ _r=1^n+1[-x̅^2x̅^νy̅^r/√(1+x̅^2)]V_k^r(h(x̅,y̅)),     for ν = 1, ...,m+1



    [(f_∗ V_k)(x̅, y̅)]^ν   =   ∑ _j=1^m+1[x̅^jy̅^ν-m-1/√(1+x̅^2)]V_k^j(h(x̅,y̅))
           +∑ _r=1^n+1[√(1+x̅^2)δ _r^ν -m-1-x̅^2y̅^ry̅^ν -m-1/√(1+x̅^2)]V_k^r(h(x̅,y̅)),

for ν =m+2, ..., m+n+2.


From the expressions of the components [(f_∗ V_k)(x̅, y̅)]^ν of the vector fields f_∗ V_k on the region ℝ^m+1× S^n, i.e. Equation (<ref>) and (<ref>), it is clear that the vector fields f_∗ V_k (k=1,...,d) are C^2-differentiable on that region. 

Now we consider the expression of the components [(f_∗ V_k)(x̅, y̅)]^ν (i.e. Equation (<ref>) and (<ref>)) of the vector fields f_∗ V_k on the other region of M^m+n+1_h_1,h_2. From Equation (<ref>) and (<ref>)), the differentiability of the components [(f_∗ V_k)(x̅, y̅)]^ν clearly depend on the differentiability of the komponents V_k^ν∘ h as well as on the differentiability of their koefisiens appearing in Equation (<ref>) and (<ref>)). Taking second partial derivative of V_k^ν∘ h with respect to x̅^j and x̅^l yields

    ∂ ^2V_k^ν (h(x̅,y̅))/∂x̅^j∂x̅^l    =    ∑ _i,a=1^m+1∂ ^2V_k^ν/∂γ ^a∂γ ^i∂γ ^i/∂x̅^j∂γ ^a/∂x̅^l
    +∑ _i=1^m+1∂ V_k^ν/∂γ ^i∂ ^2γ ^i/∂x̅^j∂x̅^l
        =    1/y̅^2+1∑ _i,a=1^m+1∂ ^2V_k^ν/∂γ ^a∂γ ^i∂ [h_1^-1(x̅)]^a/∂x̅^l∂ [h_1^-1(x̅)]^i/∂x̅^j+1/√(y̅^2+1)∑ _i=1^m+1∂ V_k^ν/∂γ ^i∂ ^2[h_1^-1(x̅)]^i/∂x̅^j∂x̅^l.

It is clear that the differentiability of V_k^ν∘ h with respect to x̅ depends only on the differentiability of V_k^ν with respect to γ, since h_1 is a diffeomorphism. Now taking second partial derivative of V_k^ν∘ h with respect to y̅^j and x̅^l yields

    ∂ ^2V_k^ν (h(x̅,y̅))/∂y̅^c∂x̅^l    =    ∑ _i,a=1^m+1∂ ^2V_k^ν/∂γ ^a∂γ ^i∂γ ^i/∂y̅^c∂γ ^a/∂x̅^l+∑ _i=1^m+1∂ V_k^ν/∂γ ^i∂ ^2γ ^i/∂y̅^c∂x̅^l+∑ _a=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^a∂κ ^r∂κ ^r/∂y̅^c∂γ ^a/∂x̅^l
        =      -1/(y̅^2+1)^2∑ _i,a=1^m+1∂ ^2V_k^ν/∂γ ^a∂γ ^i∂ [h_1^-1(x̅)]^a/∂x̅^l[h_1^-1(x̅)]^iy̅^c
           -1/(y̅^2+1)^3/2∑ _i=1^m+1∂ V_k^ν/∂γ ^i∂ [h_1^-1(x̅)]^i/∂x̅^ly̅^c
          -1/(y̅^2+1)^2∑ _a=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^a∂κ ^ry̅[h_2^-1(ŷ̅̂)]^r ∂ [h_1^-1(x̅)]^a/∂x̅^l
          +1/y̅^2+1∑ _a=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^a∂κ ^ry̅^c/y̅[h_2^-1(ŷ̅̂)]^r ∂ [h_1^-1(x̅)]^a/∂x̅^l
          -1/y̅^2+1∑ _a=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^a∂κ ^r∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^sy̅^sy̅^c/y̅^2∂ [h_1^-1(x̅)]^a/∂x̅^l
          +1/y̅^2+1∑ _a=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^a∂κ ^r∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^c∂ [h_1^-1(x̅)]^a/∂x̅^l.

It is also clear from the last equation that the second partial derivative of V_k^ν∘ h (ν ) with respect to x̅ and y̅ is continuous only if the function V_k^ν is at least C^2. The second derivative of V_k^ν∘ h with respect to y̅ is given by


    ∂ ^2V_k^ν (h(x̅,y̅))/∂y̅^c∂y̅^d    =    ∑ _i,j=1^m+1∂ ^2V_k^ν/∂γ ^j∂γ ^i∂γ ^i/∂y̅^c∂γ ^j/∂y̅^d+∑ _i=1^m+1∂ V_k^ν/∂γ ^i∂ ^2γ ^i/∂y̅^c∂y̅^d+∑ _j=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^j∂κ ^r∂κ ^r/∂y̅^c∂γ ^j/∂y̅^d
           +∑ _s,r=1^n+1∂ ^2V_k^ν/∂κ ^r∂κ ^s∂κ ^r/∂y̅^c∂κ ^s/∂y̅^d+∑ _j=1^m+1∑ _r=1^n+1∂ ^2V_k^ν/∂γ ^j∂κ ^r∂κ ^r/∂y̅^d∂γ ^j/∂y̅^c+∑ _i=1^m+1∂ V_k^ν/∂κ ^r∂ ^2κ ^r/∂y̅^c∂y̅^d,

where

    ∂γ ^i/∂y̅^c∂γ ^j/∂y̅^d    =    [h_1^-1(x̅)]^i[h_1^-1(x̅)]^jy̅^cy̅^d/(y̅^2+1)^3,


    ∂ ^2γ ^i/∂y̅^c∂y̅^d    =    3[h_1^-1(x̅)]^iy̅^cy̅^d/(y̅^2+1)^5/2 -[h_1^-1(x̅)]^iδ ^d_c/(y̅^2+1)^3/2,


    ∂κ ^r/∂y̅^c∂γ ^j/∂y̅^d    =      -y̅[h_1^-1(x̅)]^j[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/(y̅^2+1)^3 +[h_1^-1(x̅)]^j[h_2^-1(ŷ̅̂)]^r/(y̅^2+1)^2y̅^cy̅^d/y̅
           -∑_s=1^n+1[h_1^-1(x̅)]^j/(y̅^2+1)^2∂ [h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^sy̅^dy̅^sy̅^c/y̅^2 +[h_1^-1(x̅)]^jy̅^d/(y̅^2+1)^2∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^c,


    ∂κ ^r/∂y̅^d∂γ ^j/∂y̅^c    =      -y̅[h_1^-1(x̅)]^j[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/(y̅^2+1)^3 +[h_1^-1(x̅)]^j[h_2^-1(ŷ̅̂)]^r/(y̅^2+1)^2y̅^cy̅^d/y̅
           -∑_s=1^n+1[h_1^-1(x̅)]^j/(y̅^2+1)^2∂ [h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^sy̅^dy̅^sy̅^c/y̅^2 +[h_1^-1(x̅)]^jy̅^c/(y̅^2+1)^2∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^d,



    ∂κ ^r/∂y̅^c∂κ ^s/∂y̅^d   =   y̅^2[h_2^-1(ŷ̅̂)]^r[h_2^-1(ŷ̅̂)]^sy̅^cy̅^d/(y̅^2+1)^3
    +[h_2^-1(ŷ̅̂)]^r[h_2^-1(ŷ̅̂)]^sy̅^cy̅^d/(y̅^2+1)y̅^2
          
    +1/y̅^2+1∂[h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^c∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^d
          
    +1/y̅^2+1∑ _t,u=1^n+1∂[h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^t∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^uy̅^uy̅^ty̅^cy̅^d/y̅^4
          
    -2[h_2^-1(ŷ̅̂)]^r[h_2^-1(ŷ̅̂)]^sy̅^cy̅^d/(y̅^2+1)^2
    -y̅[h_2^-1(ŷ̅̂)]^ry̅^c/(y̅^2+1)^2∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^d
          
    +[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/(y̅^2+1)^2y̅∑ _t=1^n+1∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^ty̅^t
    +[h_2^-1(ŷ̅̂)]^ry̅^c/(y̅^2+1)y̅∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^d
          
    -[h_2^-1(ŷ̅̂)]^r/y̅^2+1∑ _t=1^n+1∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^ty̅^ty̅^cy̅^d/y̅^3
    +[h_2^-1(ŷ̅̂)]^sy̅^d/(y̅^2+1)^2y̅∂[h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^c
          
    -1/y̅^2+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^c∑ _t=1^n+1∂[h_2^-1(ŷ̅̂)]^s /∂ŷ̅̂^ty̅^ty̅^d/y̅^2
          
    -1/y̅^2+1∂ [h_2^-1(ŷ̅̂)]^s/∂ŷ̅̂^d∑ _t=1^n+1∂[h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^ty̅^ty̅^c/y̅^2
          
    +[h_2^-1(ŷ̅̂)]^s/(y̅^2+1)^2∑ _t=1^n+1∂[h_2^-1(ŷ̅̂)]^r /∂ŷ̅̂^ty̅^ty̅^cy̅^d/y̅^3,

and

    ∂ ^2κ ^r/∂y̅^c∂y̅^d   =   3y̅[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/(y̅^2+1)^5/2
    -y̅^c/(y̅^2+1)^3/2∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^d
          
    +1/(y̅^2+1)^3/2∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^sy̅^cy̅^sy̅^d/y̅^2
    -y̅[h_2^-1(ŷ̅̂)]^rδ ^c_d/(y̅^2+1)^3/2
           -[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/(y̅^2+1)^3/2y̅
    -[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/(y̅^2+1)^3/2y̅
    +[h_2^-1(ŷ̅̂)]^rδ ^c_d/y̅√(y̅^2+1)
          
    +1/√(y̅^2+1)∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^dy̅^c/y̅^2
    -1/√(y̅^2+1)∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^sy̅^cy̅^sy̅^d/y̅^4
           
    -[h_2^-1(ŷ̅̂)]^ry̅^cy̅^d/y̅^3√(y̅^2+1)
    -1/(y̅^2+1)^3/2∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^cy̅^d/y̅
          
    +1/(y̅^2+1)^3/2∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^sy̅^sy̅^cy̅^d/y̅^3
          
    +1/√(y̅^2+1)∂ ^2[h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^d∂ŷ̅̂^c1/y̅
    -1/√(y̅^2+1)∑ _s=1^n+1∂ ^2[h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^s∂ŷ̅̂^cy̅^sy̅^d/y̅^3
          
    +1/(y̅^2+1)^3/2∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^sy̅^sy̅^cy̅^d/y̅^2
          
    +2/√(y̅^2+1))∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^sy̅^sy̅^cy̅^d/y̅^4
           
    -δ ^c_d/√(y̅^2+1))∑ _s=1^n+1∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^sy̅^s/y̅^2
    -1/√(y̅^2+1))∂ [h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^dy̅^c/y̅^2
          -1/√(y̅^2+1)∑ _s=1^n+1∂ ^2[h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^d∂ŷ̅̂^sy̅^sy̅^c/y̅^3
          
    -1/√(y̅^2+1)∑ _s,t=1^n+1∂ ^2[h_2^-1(ŷ̅̂)]^r/∂ŷ̅̂^t∂ŷ̅̂^sy̅^ty̅^sy̅^cy̅^d/y̅^5.


All expresions appearing in Equation (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) are continuous. However, the continuity of the expresion appearing in Equation (<ref>) very depends on the form of the diffeomorphism h_2. By calculating the second derivative of y̅[h_2^-1(y̅/y̅)]^r with respect to y̅^c and y̅^d (c,d=1,2⋯, n+1) it is clear that if y̅[h_2^-1(y̅/y̅)]^r is C^2-differentiable with respect to y̅ on the region S^m×ℝ^n+1, then the expression appearing in Equation (<ref>) continuous on the region and therefore all functions V_k^ν∘ h are C^2-differentiable. By inspecting the coefisiens of the functions V_k^ν∘ h in Equation (<ref>), it easy to see that the components [(f_∗ V_k)]^ν (ν =1,2, ⋯ ,m+1) of the vector fields [(f_∗ V_k)] are C^2-differentiable on S^m×ℝ^n+1 if y̅[h_2^-1(y̅/y̅)]^r (r=1,2, ⋯ n+1) are C^2-differentiable with respect to y̅ on the region. By direct calculation, it can be concluded that if the factors [h_2^-1(y̅/y̅)]^r are C^2-differentiable with respect to y̅ for all r=1,⋯ ,n+1, then y̅[h_2^-1(y̅/y̅)]^r and y̅^s[h_2^-1(y̅/y̅)]^r, for all r,s=1,⋯ ,n+1, are also C^2-differentiable with respect to y̅. Furthermore, by inspecting the coefisiens of V_k^ν∘ h in Equation (<ref>), the components [(f_∗ V_k)]^ν (ν =m+2, ⋯ ,m+n+2) of the vector fields [(f_∗ V_k)] are C^2-differentiable on the region S^m×ℝ^n+1 if the factors [h_2^-1(y̅/y̅)]^r are C^2-differentiable on the region for all r=1,2, ⋯ n+1.



Now, we can extend the map f:S^m+n+1_s→ M^m+n+1_h_1,h_2 defined in Equation (<ref>) to a homeomorphism f_ex from an open subset U containing S^m+n+1_s of ℝ^m+n+2 onto an open subsets U' containing M^m+n+1_h_1,h_2 of ℝ^m+n+2. Let ι_S be the natural inclusion of S^m+n+1_s into ℝ^m+n+2 and G a Riemannian metric tensor in the ambient space ℝ^m+n+2. Furthermore, let g denote the Riemannian metric tensor on S^m+n+1_s inherited from G, i.e. g=ι _S^∗ G. By using the map h_ex, i.e. the invers of f_ex, we can pull back the metric tensor field G yielding the Riemannian metric tensor h_ex^∗G in U'. Let ι _M be the natural inclusion of M^m+n+1_h_1,h_2 into ℝ^m+n+2. Then, the pull back ι _M^∗ h_ex^∗G of h_ex^∗G by ι _M is a Riemannian tensor metric on M^m+n+1_h_1,h_2. It is easy to show that the metric tensor ι _M^∗ h_ex^∗G is the metric tensor h^∗ g=h^∗ (ι _S^∗ G). The metric tensor ι _M^∗ h_ex^∗G can be regarded as the metric tensor g on S^m+n+1 when it is described with the exotic differential structure. The metric tensor field g gives rise to the Riemannian measures ν on S^m+n+1_s and the metric tensor field h^∗ g to the Riemannian measure h^∗ν on M^m+n+1_h_1,h_2 respectivelly. 
Now define

    |f_∗ X((q̅))|_h^∗ g:=(h^∗ g)(q̅)((f_∗ X)(q̅),(f_∗ X)(q̅)),

for every vector field X defined on M^m+n+1_h_1,h_2. It is straightforward to show that

    |f_∗ X((q̅))|_h^∗ g=|X|_g(h(q̅)).

Since all the elements of the matrix representing the pull-back map h^∗ as well as of the push-forward map f_∗ are continuous then

    ∫ _M^m+n+1_h_1,h_2|f_∗ V_0((q̅))|^p(h^∗ν ) (dq̅)<∞,

whenever

    ∫ _S^m+n+1_s|V_0(q)|^pν (dq)<∞,

for 1≥ p, i.e. f_∗ V_0∈ L^p(TM^m+n+1_h_1,h_2).


Since V_0∈ℍ^p_1(TS^m+n+1_s), then there exist m+n+1 vector fields Y_1,...,Y_m+n+1 on S^m+n+1_s so that for σ = 1,⋯ , m+n+1 we have

    ∫ _S^m+n+1_s∂φ/∂ q^σV_0(q)ν (dq)=-∫ _S^m+n+1_sφ (q)Y_σ (q)ν (dq),

(the integrations appearing in Equation (<ref>) are in the Bochner sense) for every φ∈ C_c^∞ (S^m+n+1_s) and

    ∫ _S^m+n+1_s|∇ V_0 (q)|_g^pν (dq)<∞ ,

where

    |∇ V_0(q)|_g=(∑ _η =1^m+n+1|Y_η (q)|_g^2)^1/2=(∑ _η =1^m+n+1g(q)(Y_η (q),Y_η (q)
    ))^1/2.

From Equation (<ref>) we obtain

    |f_∗∇ V_0(q̅)|_h^∗ g=(∑ _η =1^m+n+1(h^∗ g)(q̅)(f_∗ Y_η (q̅),f_∗ Y_η (q̅)
    ))^1/2=|∇ V_0(q̅)|_g(h(q̅)).

Since all the elements of the matrix representing the pull-back map h^∗ are continuous, then

    ∫ _M^m+n+1_h_1h_2|f_∗∇ V_0 (q̅)|^p(h^∗ν )(dq̅)<∞,

and therefore f_∗ V_0 is contained in the Sobolev space ℍ^p_1(TM^m+n+1_h_1,h_2).


Furthermore, if V_0 is in the space L^∞ (TS_s^m+n+1), then f_∗ V_0 belongs to the space L^∞ (TM^m+n+1_h_1,h_2) of all bounded measurable vector fields on M^m+n+1_h_1,h_2, since the map f_∗ is continuous. Now because S^m+n+1_s as well as M^m+n+1_h_1,h_2 is compact and V_0∈ L^∞ (TS_s^m+n+1), it is clear that V_0 is in L^1 (TS_s^m+n+1) and f_∗ V_0 is in L^1(TM^m+n+1_h_1,h_2). If V_0 has (weak) divergence with div V_0∈ L^∞ (S^m+n+1_s), then so is f_∗ V_0 with div (f_∗ V_0)∈ L^∞ (M^m+n+1_h_1,h_2), since the maps f_∗ and h^∗ are continuous.


As a conclusion we have

	Let M^m+n+1_h_1,h_2 be an exotic sphere, where h_2^-1(y̅/y̅) is C^2-differentiable with respect to y̅. Furthermore, let V_0∈ L^∞ (TS^m+n+1_s)∩ℍ^p_1(TS^m+n+1_s) for some p≥ 1, div  V_0∈ L^∞ (S^m+n+1_s), and for each k=1,⋯ ,d, V_k∈ C^2(TS^m+n+1_s) so that q_t(q) is the unique ν-almost everywhere stochastic invertible flow of (<ref>). Then, the flow (f∘ q_t∘ h)(q̅) on M^m+n+1_h_1,h_2 is the unique h^∗ν-almost everywhere stochastic invertible flow of (<ref>) in the sense of Definition <ref>


With the same conditions on the vector fields V_k (k=0,1,⋯,d), Theorem <ref> means that whenever h_2^-1(y̅/y̅) is C^2-differentiable with respect to y̅, both stochastic flows q_t(q) and (f∘ q_t∘ h)(q̅) have the same regularities. It also means, both description (using the standard diferential structure as well as using the choosen exotic differential structure) of the stochastic flow on the sphere S^m+n+1 look the same or indistinguishable. If both description of the stochastic flow on the sphere S^m+n+1 are distinguishable, then the diffeomorphism h_2^-1(y̅/y̅) is not C^2-differentiable with respect to y̅. Now if the map h_2^-1(y̅/y̅) is not C^2-differentiable with respect to y̅, then both description of the stochastic flow on the sphere S^m+n+1 may or may not indistinguishable.

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