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I do not know if the following question makes sense.

Is it possible to define an inner product (that gives real values) for vectors on a sphere $S^n$ (let say $S^1$)? The set of these vectors is not a real vector space but probably we can handle somehow to create an inner product.

Do you know any example?

Edit:

I define a map on a real vector space (say $\mathbb{R}^n$) and I would like to prove that it is indeed an inner product to use the Cauchy-Schwarz inequality. But it fails to satisfy the definite-positiveness over the whole vector space. However, restricted on a sphere of this vector space, the definite-positiveness holds. But in this case, let $v$ be a vector in the sphere, a vector $\alpha v$ for $\alpha \in \mathbb{R}^n$ is not in the sphere. So the inner-product is not well-defined. How I get rid of that?

Besides, the only things I want is the use of Cauchy-Schwarz inequality - the property "like an inner product". Do you know something related?

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 Manifolds have metrics: smoothly varying family of inner products on each of its tangent spaces. Is this what you want? At any rate, this is probably not an appropriate question for this site. Perhaps you can ask in one of the places mentioned in the FAQ. – José Figueroa-O'Farrill Apr 7 2011 at 9:15
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 Without knowing what existing properties of an inner product you want to preserve, this question doesn't seem well-defined. – Yemon Choi Apr 7 2011 at 9:37
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 That is: if you want to combine two vectors on your sphere to get a number, then there are many different ways to do that. What other properties that are "like an inner product" are you looking for? – Yemon Choi Apr 7 2011 at 9:38
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 What's wrong with the restriction of the usual inner product on $\mathbb R^{n+1}$ to $S^n$? – Emil Jeřábek Apr 7 2011 at 11:45
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 You did not defined the problem, but maybe these possibilities can help you. If not, you can try to define the problem better. A) If by "vectors on a sphere $S^n$" you mean restrictions of the vectors from $\mathbb R^{n-1}$, you can apply the solution of Emil Jeřábek - the restriction of the inner products on $\mathbb R^{n-1}$. B) If by "vectors on a sphere $S^n$" you mean tangent vectors to the sphere, take the inner product of tangent vector given by the metric which is the restriction of the natural metric on $\mathbb R^{n-1}$. Here I see two subcases: – Cristi Stoica Apr 7 2011 at 12:34
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closed as too localized by José Figueroa-O'Farrill, Bill Johnson, Andreas Thom, Andrew Stacey, Simon Thomas Apr 7 2011 at 12:31

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