METRIC GEOMETRY OF CARNOT-CARATH´ EODORY SPACES WITH C 1 -SMOOTH VECTOR FIELDS Sergey Vodopyanov Bia � l owieˆ za, Poland, XXXI Workshop on Geometric Methods in Physics 24 June – 30 June, 2012
REFERENCES 1. S. K. Vodopyanov and M. B. Karmanova, “Local approxima- tion theorem on Carnot manifolds under minimal smoothness”, . Dokl. AN , 427, No. 6, 731–736 (2009). 2. M. Karmanova and S. Vodopyanov, “Geometry of Carnot– Carath´ eodory spaces, differentiability and coarea formula”, Anal- ysis and Mathematical Physics , Birkh¨ auser, 284–387 (2009). 3. M. Karmanova “A new approach to investigation of Carnot– Carath´ eodory geometry", GAFA 21 , no. 6 (2011), 1358–1374. 4. A. V. Greshnov, “A proof of Gromov Theorem on homo- geneous nilpotent approximation for C 1 -smooth vector fields”, Mathematicheskie Trudy , 15, No. 2 (2012). 5. S. G. Basalaev and S. K. Vodopyanov, “Approximate differen- tiability of mappings of Carnot–Carath´ eodory spaces”, Eurasian Math. J. , 3 (2012): arXiv:1206.5197.
• Mathematical foundation of thermodynamics • Carnot, Joules There exist thermodynamic states A , B that can not be connected to each other by "adiabatic process". This impossibility is related to the impossibility of perpetual mo- tion machines. • 1909, Carath´ eodory proving the existence of entropy derived the following statement: Let M be a connected manifold endowed with a corank one dis- tribution. If there exist two points that can not be connected by a horizontal path then the distribution is integrable. It is a solution of dE + PdV = TdS.
• Mathematical foundation of thermodynamics • Carnot, Joules There exist thermodynamic states A , B that can not be connected to each other by "adiabatic process". This impossibility is related to the impossibility of perpetual mo- tion machines. • 1909, Carath´ eodory in order to prove the existence of entropy derived the following statement: Let M be a connected manifold endowed with a corank one dis- tribution. If there exist two points that can not be connected by a horizontal path then the distribution is integrable. It is a solution of dE + PdV = TdS.
• Development • Carath´ eodory 1909, Rashevskiy 1938, Chow 1939: arbitrary two points of M can be joined by a “horizontal” curve. It follows that ( M , d c ) is a metric space with the subriemannian distance d c ( u, v ) = inf { L ( γ ) | γ is horizontal, γ (0) = u, γ (1) = v } not comparable to Riemannian one.
• Development • Carath´ eodory 1909, Rashevskiy 1938, Chow 1939: arbitrary two points of M can be joined by a “horizontal” curve. It follows that ( M , d c ) is a metric space with the subriemannian distance d c ( u, v ) = inf { L ( γ ) | γ is horizontal, γ (0) = u, γ (1) = v } not comparable to Riemannian one.
• H¨ ormander, 1967: Hypoelliptic equations A problem: when a distribution solution f to the equation ( X 2 1 + . . . + X 2 n − 1 − X n ) f = ϕ ∈ C ∞ is a smooth function? Here X i ∈ C ∞ . • Particular case: ? Kolmogorov’s equations ∂ 2 u ∂x 2 + x∂u ∂y − ∂u ∂t = f • physics (diffusion process), economics (arbitrage theory, some stochastic volatility models of European options), etc.
• H¨ ormander, 1967: Hypoelliptic equations A problem: when a distribution solution f to the equation ( X 2 1 + . . . + X 2 n − 1 − X n ) f = ϕ ∈ C ∞ is a smooth function? Here X i ∈ C ∞ . • Particular case: Kolmogorov’s equations ∂ 2 u ∂x 2 + x∂u ∂y − ∂u ∂t = f • physics (diffusion process), economics (arbitrage theory, some stochastic volatility models of European options), etc.
Hypoelliptic Equations • H¨ ormander (1967) : sufficient conditions on fields X 1 , . . . , X n : There exists M < ∞ such that • Lie { X 1 , X 2 , . . . , X n } = span { X I ( v ) | | I | ≤ M } = T v M for all v ∈ M where X I ( v ) = span { [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ] . . . ]( v ) : X i j ∈ H 1 } for I = ( i 1 , i 2 , . . . , i k ) . • M is the depth of the sub-Riemannian space M . • Stein (1971) : The program of studying of geometry of H¨ ormander vector fields; description of singularities of fundamental solutions
Hypoelliptic Equations • H¨ ormander (1967) : sufficient conditions on fields X 1 , . . . , X n : There exists M < ∞ such that • Lie { X 1 , X 2 , . . . , X n } = span { X I ( v ) | | I | ≤ M } = T v M for all v ∈ M where X I ( v ) = span { [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ] . . . ]( v ) : X i j ∈ H 1 } for I = ( i 1 , i 2 , . . . , i k ) . • M is the depth of the sub-Riemannian space M . • Stein (1971) : The program of studying of geometry of H¨ ormander vector fields; description of singularities of fundamental solutions
Hypoelliptic Equations • H¨ ormander (1967) : sufficient conditions on fields X 1 , . . . , X n : There exists M < ∞ such that • Lie { X 1 , X 2 , . . . , X n } = span { X I ( v ) | | I | ≤ M } = T v M for all v ∈ M where X I ( v ) = span { [ X i 1 , [ X i 2 , . . . , [ X i k − 1 , X i k ] . . . ]( v ) : X i j ∈ H 1 } for I = ( i 1 , i 2 , . . . , i k ) . • M is called the depth of the sub-Riemannian space M . • Stein (1971) : The program of studying of geometry of H¨ ormander vector fields; description of singularities of fundamental solutions
Geometric control theory ⋄ The linear system of ODE ( x ∈ M N , m < N ) . n � X i ∈ C ∞ . (1) x = ˙ a i ( t ) X i ( x ) , i =1 • Problem: To find measurable functions a i ( t ) such that system (5) has a solution with the initial data x (0) = p , x (1) = q . If system (5) has a solution for every q ∈ U ( p ) then it is called locally controllable. It is locally controllable iff Lie { X 1 , X 2 , . . . , X n } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X n } is bracket-generating.
Geometric control theory ⋄ The linear system of ODE ( x ∈ M N , n < N ) . n � X i ∈ C ∞ . x = ˙ a i ( t ) X i ( x ) , (2) i =1 • Problem: To find bounded measurable functions a i ( t ) such that system (5) has a solution with the initial data x (0) = p , x (1) = q . If system (5) has a solution for every q ∈ U ( p ) then it is called locally controllable. It is locally controllable iff Lie { X 1 , X 2 , . . . , X n } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X n } is bracket-generating.
Geometric control theory ⋄ The linear system of ODE ( x ∈ M N , m < N ) . n � X i ∈ C ∞ . x = ˙ a i ( t ) X i ( x ) , (3) i =1 • Problem: To find measurable functions a i ( t ) such that system (5) has a solution with the initial data x (0) = p , x (1) = q . If system (5) has a solution for every q ∈ U ( p ) then it is called locally controllable. It is locally controllable iff Lie { X 1 , X 2 , . . . , X n } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X n } is bracket-generating.
Geometric control theory ⋄ The linear system of ODE ( x ∈ M N , n < N ) . n � X i ∈ C ∞ . x = ˙ a i ( t ) X i ( x ) , (4) i =1 • Problem: To find measurable functions a i ( t ) such that system (5) has a solution with the initial data x (0) = p , x (1) = q . If system (5) has a solution for every q ∈ U ( p ) then it is called locally controllable. • It is locally controllable iff Lie { X 1 , X 2 , . . . , X n } = T M , i.e. the “horizontal” distribution H M = { X 1 , X 2 , . . . , X n } is bracket-generating.
APPLICATIONS of SUBRIEMANNIAN GEOMETRY • Thermodynamics • Non-holonomic mechanics • Geometric Control Theory • Subelliptic equation • Geometric measure theory • Quasiconformal analysis • Analysis on metric spaces • Contact geometry • Complex variable • Economics • Transport problem • Quantum control • Neurobiology • Tomography • Robotecnics
eodory space ( C 1 -smooth vector fields) Carnot–Carath´ • M is a connected C ∞ -smooth manifold with dim top ( M ) = N ; в T M существует фильтрация подрасслоениями H M = H 1 M � . . . � H i M � . . . � H M M = T M H i M ( v ) = span { X 1 ( v ) , . . . , X dim H i ( v ) } , dim H i M ( v ) = dim H i ; � [ X i , X j ]( v ) = c ijk ( v ) X k ( v ) , k : deg X k ≤ deg X i +deg X j где deg X k = min { m : X k ∈ H m }
eodory space ( C 1 -smooth vector fields) Carnot–Carath´ • M is a connected C ∞ -smooth manifold with dim top ( M ) = N ; • in T M there exists a filtration by subbundles H M = H 1 M � . . . � H i M � . . . � H M M = T M ; � [ X i , X j ]( v ) = c ijk ( v ) X k ( v ) , k : deg X k ≤ deg X i +deg X j где deg X k = min { m : X k ∈ H m }
eodory space ( C 1 -smooth vector fields) Carnot–Carath´ • M is a connected C ∞ -smooth manifold with dim top ( M ) = N ; • in T M there exists a filtration by subbundles H M = H 1 M � . . . � H i M � . . . � H M M = T M ; • ∀ v ∈ M ∃ U ( v ) and vector fields X 1 , X 2 , . . . , X N ∈ C 1 such that H i M ( v ) = span { X 1 ( v ) , . . . , X dim H i ( v ) } , dim H i M ( v ) = dim H i ; � [ X i , X j ]( v ) = c ijk ( v ) X k ( v ) , k : deg X k ≤ deg X i +deg X j где deg X k = min { m : X k ∈ H m }
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