HOMEWORK MATH-GA 2350.001 DIFFERENTIAL GEOMETRY I (1, 2, 4 are due by September, 19, 2016, 3,5, 6 are due by September, 26, 2016)

1. Show that a topological manifold M is connected iff M is path-connected.

2. Let RP^{n} be the n-dimensional projective space, with the atlas given by the
following functions

φ_{i} :U_{i} →R^{n},[x_{1}, . . . x_{n+1}]7→(x_{1}

x_{i}, . . . ,xi−1

x_{i} ,x_{i+1}

x_{i} , . . .x_{n+1}
x_{i} ),

whereU_{i}, i= 1, . . . n+ 1 are open subsets{[x_{1}, . . . , x_{n+1}], x_{i} 6= 0}.

(a) Show that (U_{i}, φ_{i}) is a smooth atlas.

(b) Show that RP^{1} is diffeomorphic to S^{1}.

(c) Let π : S^{2} → RP^{2} be a map that sends a point (x, y, z) to a unique
line through this point. Show that π is smooth and that π is local
diffeomorphism: for any pointp∈S^{2} there exists an open neighborhood
U ⊂M such that π_{U} :U →π(U) is a diffeomorphism on an open subset
of RP^{2}.

3. Let0< r≤m≤n. LetV_{r} ⊂Mn×m(R)be the set of matrices of rankr. Show
that Vr is a smooth submanifold ofMn×m(R) and compute its dimension.

4. Let M be a manifold of class C^{k}. Let A, B ⊂M be closed subsets such that
A∩B =∅. Show that there is a functionf ∈C^{k}(M)with values in [0,1]and
such thatf is identically0 onA and identically 1 onB.

5. Is product of two smooth manifolds with boundary a smooth manifold with boundary?

6. Let n > 0 be an integer and leth,i be the euclidean scalar product on R^{n+1},
and S^{n} := {x ∈ R^{n+1},||x|| = 1}. If x ∈ R^{n+1} \ 0, we denote by [x] the
corresponding point ofRP^{n}.

(a) Show that the mapf :S^{n}×S^{n}→Rdefined by(x, y)7→ hx, yiis smooth.

Find all points were it is a submersion.

(b) Let M ⊂ S^{n} ×S^{n} consists of orthogonal couples. Show that M is a
smooth submanifold^{1} of S^{n}×S^{n}.

(c) Let M^{0} ⊂ RP^{n}×RP^{n} consists of couples of orthogonal lines (L, L^{0}) of
R^{n+1}. Show that M^{0} is a smooth submanifold ofRP^{n}×RP^{n}.

(d) LetE be the set of triples(x, x^{0}, y)ofS^{n}×S^{n}×R^{n+1} such thathx, x^{0}i=
hx, yi=hx^{0}, yi= 0 andπ:E →M be the map(x, x^{0}, y)7→(x, x^{0}). Show
that π is a smooth vector bundle over M.

1A subsetM ⊂N of a smoothn-manifoldN is a smooth submanifold if for any pointp∈M
there is a chart(U, φ)ofN atpsuch thatφ(U∩M)is a submanifold ofφ(U)⊂R^{n} in the sense
of one of the four equivalent definitions given during the lecture

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(e) Let E^{0} be the set of couples ([x], y) of RP^{n}×R^{n+1} such that hx, yi= 0
and π^{0} :E^{0} →RP^{n} be the map ([x], y)7→ [x]. Show that π^{0} is a smooth
vector bundle.

(f) Let E^{00} be the set of couples (([x], y),([x^{0}], y^{0})) of E^{0} × E^{0} such that
hx, x^{0}i = 0 and y = y^{0} and let π^{00} : E^{00} → M be the map defined by
(([x], y),([x^{0}], y^{0})) 7→ ([x],[x^{0}]). Show that π^{00} is a is a smooth vector
bundle.

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