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• 19.3k
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### Example of "giving" a metric to an intrinsically defined manifold

Here is a nice example: complex projective space $\mathbb{CP}^n$ is a space parameterizing $1$-dimensional complex subspaces of $\mathbb{C}^{n+1}$. It turns out to be a manifold, and it can be ...
• 379k
1 vote
Accepted

### Simple computation on the identity on the definition of Donaldson-Futaki invariant

It seems there is a mistake in the degrees of the polynomials, in your computation. Notice that $w_k$ is of degree $n+1$, while $d_k$ is of degree $d$. So, the two polynomials in the numerator and ...
• 1,047
1 vote

### About closed curves in a dynamical systems

Here's a sketch for a solution that might work. Look at a simpler case first: that is for a circle $S^1$. Suppose we have a $C^1$ flow defined on $S^1$. We can express this as an ODE using polar ...
• 1,488
1 vote
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### Differential structure on $\mathbb S^n/±\mathrm{id}$ such that the injection $i:\mathbb RP^n\rightarrow\mathbb S^n/(±\mathrm{id}$ is a diffeomorphism

As you say, the the natural injection $i:\mathbb S^n\rightarrow\mathbb R^{n+1}\backslash\{0\}$ induces a continuous map $i': \mathbb S^n/±\mathrm{id} \to \mathbb RP^n$ on the quotient spaces. That $i'$...
• 61.3k

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