feedbackform OHandpreregform Assignments postedon friday marital Pointset topology TopologicalManifold Smooth Riemannian manifold A dim Manifolds in IR MAR MEIR is a K dim manifold in if APE M FU neighhdofp and HEIR VER Lapiece DIRK that isopen and a function f U satisfying Thatisdeformed 1 fisc Recall 2 V IS a 2 fisa homeomorphism f 3 Df ish rank a jIIHpf UES tpqpfmxox.FIsetfallnectors starting U 7 DAIRY fCq P Tps PM Image f Rfy s TqV i Dfw 2 g SEIR dim 2 Since Dfisl l the tangentspace is welldefined andis Jthe samedimension TTT Remake We want K dim manifolds in IR to satisfy the more general def that we are trying to come up with Point Set To p o l o g y Let M be a set Need the notionofcontinuity compact neighbourhood connected a I hit in an a these notions To p o l o g i c a l structure first Notice i Continuity in IRA could be formulated only using A opensets Let T E PCM satisfying i il Q M ET 2 T isclosed under unions and finite intersections If so we say MT is a topological space with topology T Appendix Reimann Read A mandatory To p o l o g i c a l Properties 1 closed AEM is closed it A isopen 2 Compact 3 connectedness 4 continuous i f i M M i cont if f A is open whenever A isopen 5 Bounded Not atopology property It depends on the fact Ad PhxRn R Rq tellfqH 6 differentiable function Not a topological property It depends on V S ofIRN We don'thave a notion ofaddition subtraction Reina A property istopological iff it stays inramat under homeomorphisms topological invariant homeomorphism A e B Subspace To p o lo g y If M T is a topological space then AEM inherits a topology from the ambient space Characterized by Unique topology s t c t i finish where i A M i pl p let Ta Anu UET TA is called the subspace topology exc i Ven b TA is a topology on A Based Ded Let MT be a topological space BET is called a basis for T if forevery VET andPEO 3 BED s t PE B E U equivalently Bisabasis for T if every VET is a union of elements in B Rn EI RM B Bros PE reft topology is a basis for the regular In fact we can find a countable basis B Berle Pean real Defra Mrt is second countable if it admits a countable basis Proposition A subspace A a secondcountablespace is also second countable Def Mrt is a Hausdorff space if thqEM peg 7 disjoint open sets and V s t PE U and qe V Proposition Any subspace of a Hausdorff spaceis Hausdorff Hausdorff Proposition x one closed in spaces a MEAD open Backto Manifolds Topol ogical Manifolds et Mit be a topological space Suppose Mis Hausdorff andsecond countable Te c hn i c a lre a s o n g i 1 Partition ofunity 2 Compact exhaustion 3 Existenceof a Riemannianmeth 4 Embedding theorems Dedi Mit is locally Euclidean dim m if C IRM open t PEM 3 U neighbd of P and F V is a homeomorphism and 70 U V that vern fcq Cx i'm x q XmCq Xi U IR coordinates of geo IMPORTANT TERMINOLOGY Vid is called a coordinate chat COFFIN U is called a coordinateneighbnd of P is called a coordinate map coordinate system equivalent to M is locally homeomorphic to Rm DeffTopologicalManifat M is a topological manifold of dim rn if it's a Hausdorff second countable topological space that is locally Euclidean f din m Is the dimension well defined amount M alum Tait i 012 Uz zooli 01 n ERM EIR drool CU AUD RIGAUD homeomorphism theorem Invariance dimension If UEIRM and VERN are opensets that are homeomorphic then man Needs the toolsof AlgebraicTo p o l o g y beyondthecourse Definition of smooth Manifold Let M be a topological manifold ofdim n Consider two coordinate charts UN and CU 4 00 y g transitionmap n called DEIR homeomorphism g u 4am i 2 oleum n C IRn E IR Det i 2 charts Uol and CV 4 are C compatible if the transitionmap 40 is a diffeomorphism 4051 Dam Kuru anti econ acorns Det A atlas is a collection of charts manifold A Va Ola covering the and are pairwise co compatible precisely 1 Y U M 2 Varda and UB Op on copmpatible tap Question Is C compatible an equivalence relation Reflexive Symmetric transitive Suppose Ul Q is compatible with Uz oh Uz Az is compatible with 03,03 question is CU oh compatible with Us.dz more precisely i is also idiots 013 Us a diffeomorphism U U AUS Akira's Answer whereisthisthe it'sonline on 01 01300i Hopi µ Giffen 4 differ 3001 is a diffeomorphism 013cm i dillies all we can say 3001 is a chiffon Nat transitive 110 EIU fledEU EV C u Example g S uses c S ee c i 2K gotta i U Co 21T Gosa Sino O 0 Yi V C IT TI cos 0 Sino is ELTIM 0104 i 4 UN Kono diffeomorphism i show it's a a Thismakes Chol CU 413 atlas far s Detfasmathmaniff Asmooth manifold Adimn is a topological manifold of dimn equipped with a C Atlas The atlas gives thetopological manifold a Smooth structure makingit a smooth manifold.