Sina
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If you want a more rigorous book with geometric motivation I reccomend John M. Lee`s topological manifolds where he does a lot of stuff on covering spaces homologies and cohomologies. As a supplement ...

To do this computation in coordinates without using functions and points you have to adopt the physicist way of writing things which is messy and unplesant :-) However chain rule takes care of all the ...

Given a hypersurface $S$ in a manifold $M$ with volume form $\omega$ you can construct the volume element on $S$ induced by $\omega$ as follows: Take a field of unit normal vectors $N$ along $S$ and ...
If $A$ is an invertible operator of an n dimensional vector space to itself then $$\frac{1}{det(A^{-1})^{1/n}} \leq |A|\leq \frac{|A^{-1}|^{n-1}}{det(A^{-1})}$$ This you can get by the relation $$... View answer 3 votes Please see, Page 55 of Prof. Oliveira's notes: http://cdsagenda5.ictp.trieste.it/askArchive.php?base=agenda&categ=a11165&id=a11165s16t18/lecture_notes There he partially answers this ... View answer 2 votes As Ian explained above, the Wiener measure of a path is indeed 0. In general, whenever you have uncountable number of elements in your space, your probability measure can be only non-zero on a ... View answer 2 votes I was able to give a partial answer to my question, the part where I ask about the relation between linear connection on M and Ehresmann connections on TM. It goes as follows. Given a linear ... View answer 2 votes Simply because you pull back the equation \omega(X,\cdot) = dH to an equation on T^*(TM) using the Legendre transformation. Indeed denote \phi_L: TM \rightarrow T^*M the Legendre transformation ... View answer Accepted answer 2 votes a) \int f d\mu = \int_{X_a} f d\mu + \int_{(X_a)^c} f d\mu . But \int_{X_a} f d\mu > a\mu(X_a) and since f is positive \int_{(X_a)^c} f d\mu>0. Therefore \int f d\mu > a\mu(X_a) ... View answer Accepted answer 2 votes Let T = \{D \in Q^{2 \times 2} | D^t = D\} Let B \in im(\phi). Then B = A + A^t so B^t = A^t + A = B. So im(\phi) \subset T. Let A \in T so that A^t = A. Then write A = \frac{A + A}{2}... View answer 1 votes Ergodic hypothesis more or less states the the Hamiltonian flow restricted to each energy surface is ergodic for the Liouville measure. Now if you take an integrable Hamiltonian system with a 2n ... View answer 1 votes By definition a coordinate system is a collection of charts \{U_i,\phi_i\}_i where U_i are open sets in your manifold that cover your whole manifold and \phi_i:U_i \rightarrow \mathbb{R}^n are ... View answer 1 votes Okay, I found the answer for the second question in Kobayashi's book: it says isometries of a riemannian manifold (with some additional conditions) forms a lie group whose lie algebra is the space ... View answer Accepted answer 0 votes Okay, I think I got the answer. For t irrational, let t_n be an increasing sequence of rationals converging to t and define the sets C'_n = C'(t_n,U). If \gamma \in C'(t,U) (defined using ... View answer Accepted answer 0 votes Motivated by the answer above it seems it is easy to put the answer into a form (following Cartan) which relates the Frenet frame and the Darboux frame of the curve. I assume the curve C(t)i s arc ... View answer 0 votes If the limit exists then for all x_n \rightarrow 0 it must be that f(x_n) \rightarrow c where c is the limit. But for every number r in [-1,1] you can find a sequence x_n such that f(x_n) ... View answer Accepted answer 0 votes Let the left top corner of the rectangle be P5. Call a=|P5-P3| (distance between P5 and P3) and b = |P5-P1|. Then$$H^2 = b^2 + (X+a)^2A^2 = b^2 + a^2  Solve the second one for b, i.e \$b^2 = ...